Measurement and Modeling of the Directional Scatter of Light
111
Transcript of Measurement and Modeling of the Directional Scatter of Light
Measurement and Modelling of the
Directional Scattering of Light
Derek John Grith
2015
Measurement and Modelling of the
Directional Scattering of Light
by
Derek John Grith (212561361)
BSc (Physics)
Submitted in partial fullment of the requirements
for the degree of Master of Science in Physics
School of Chemistry and Physics
College of Agriculture, Engineering and Science
University of KwaZulu-Natal
Pietermaritzburg
27 February 2015
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Declaration
This thesis describes the work undertaken at the University of KwaZulu-Natal under
the supervision of Dr N. Chetty between July 2012 and February 2015.
I declare the work reported herein to be my own research, unless specically indic-
ated to the contrary in the text.
Signed: ..................................................
D. J. Grith
On this ................ day of ............................ 2015
I hereby certify that this statement is correct.
Signed: ..................................................
Dr N. Chetty (Supervisor)
On this ................ day of ............................ 2015
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Acknowledgements
My wife, Ingrid Swart, has made this project possible in many more ways than she
knows. I dedicate the result to her.
My supervisor, Naven Chetty, has played a crucial role not only in providing guid-
ance on all fronts and keeping this work moving ahead, but also in creating the original
opportunity.
I thank also my employer, CSIR, and all of my colleagues there for providing an
environment in which further study is encouraged and rewarded.
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Abstract
The quantum nature of light suggests that a photon can interact with matter in two
primary ways. Firstly and perhaps more simply, the photon could be absorbed or
secondly and more complex, it could be scattered into a new direction of propagation.
The scattering process can be thought of as probabilistic, with a statistical distribution
of possible new directions of travel with respect to the original. In the case of interaction
with a small particle of matter, the probability distribution is referred to as the phase
function. In the case of scattering at a surface interface between two bulk materials,
the new direction of travel is distributed according to a function called the Bidirectional
Scattering Distribution Function (BSDF). The BSDF depends on both the direction of
arrival and the direction of scatter (hence bidirectional), the type of material and the
condition of the surface as well as the wavelength of light.
This work explores a number of areas related to the BSDF, with special attention
to the eects of random light scatter in high performance optical imaging systems such
as space telescopes. These demanding imaging applications require optical components
manufactured to very high standards with respect to shape, smoothness and cleanliness.
This means that random scatter from the surfaces of these optical components must
be controlled to very low levels. The measurement of very weak optical surface scatter
is therefore a problem of particular interest. An interferometric technique has been
proposed here for improving the quality of such measurements. The interference eects
produced in the image by this technique were analysed using Nijboer-Zernike diraction
theory, leading to a journal publication in Current Applied Physics.
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Contents
1 Background and Motivation 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Precision Imaging Optical Systems . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Aberration Retrieval . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Theoretical Background 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Denitions and Terminology . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Bidirectional Scattering Distribution Function . . . . . . . . . . 9
2.2.2 BRDF and BTDF . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Denitions Specic to Optical Surfaces . . . . . . . . . . . . . . . . . . 11
2.3.1 Total Integrated Scatter . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Surface Topography and Power Spectral Density . . . . . . . . . 15
2.3.3 Autocovariance Function . . . . . . . . . . . . . . . . . . . . . . 16
2.3.4 Angular Resolved Scattering . . . . . . . . . . . . . . . . . . . . 16
2.3.5 Surface PSD Relationship to Image Quality . . . . . . . . . . . 17
2.4 Optical Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Rayleigh-Rice Scattering . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Generalised Harvey-Shack Scattering . . . . . . . . . . . . . . . 20
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2.5 Scatter Models for Stray Light Analysis . . . . . . . . . . . . . . . . . . 21
2.5.1 The Harvey Models . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.2 The K-Correlation Model . . . . . . . . . . . . . . . . . . . . . 23
2.5.3 Scatter Model Selection . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Surface Prolometry and BSDF . . . . . . . . . . . . . . . . . . . . . . 25
3 Measurement of Weak Scattering in the Specular Beam 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Segmented Aperture Interferometry . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Focal Plane Irradiance . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Source and Detector Aperture Eects . . . . . . . . . . . . . . . . . . . 39
3.4 Bilaterally Symmetric Segmented Pupils . . . . . . . . . . . . . . . . . 42
3.5 Circular Pupils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Imperfect Nulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6.1 Polychromaticity . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6.2 Non-Uniform Illumination or Vignetting . . . . . . . . . . . . . 50
3.6.3 Geometry Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6.4 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Precision Imaging Systems 52
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Optical Design and Opto-mechanical Design . . . . . . . . . . . . . . . 55
4.3 Optical Manufacture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Operational Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Conclusion 61
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Bibliography 62
6 Journal Paper Submission 77
A Derivations 88
A.1 Phase Delay of a Tilted Plane Parallel Plate . . . . . . . . . . . . . . . 88
B Article in Press and Acceptance Correspondence 91
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Table of Abbreviations
Abbreviation Denition
ACV Auto-Covariance function
AFM Atomic Force Microscope (or Microscopy)
ARS Angular Resolved Scattering function
BK Beckmann-Kirchho (Scattering theory)
BRDF Bidirectional Reectance Distribution Function
BSDF Bidirectional Scattering Distribution Function
BTDF Bidirectional Transmission Distribution Function
CNZ Classical Nijboer-Zernike (Theory)
DUV Deep Ultraviolet
EO Earth Observation
ENZ Extended Nijboer-Zernike (Approach or Analysis)
EUV Extreme Ultraviolet
FT Fourier Transform
FCT Fourier Cosine Transform
GHS Generalised Harvey-Shack (Scattering theory)
HS Harvey-Shack (Scattering theory)
IC Integrated Circuit
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MTF Modulation Transfer Function
NA Numerical Aperture
NSC Non-Sequential Component (Zemax®)
OPD Optical Path Dierence
OPL Optical Path Length
OTF Optical Transfer Function
PSD Power Spectral Density
PSF Point Spread Function
PTF Phase Transfer Function
RR Rayleigh-Rice (Scattering theory)
SRF Spectral Response Function
TIS Total Integrated Scatter
UV Ultraviolet
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Chapter 1
Background and Motivation
1.1 Introduction
Scattering of light from material surfaces and particles is an important eld of study
for numerous scientic and technological disciplines including computer vision/graphics
[13], radiative transfer [4, 5], optical systems [6, 7] and Earth observation by remote
sensing [810]. Major determinants of the scattered spatial light eld are the type of
material and the surface topography (prole, shape, texture, roughness, structure) [11].
The intensity of light scattered in dierent outgoing directions varies with both the light
wavelength and the directions from which the light arrives at the surface or particle
[11,12].
When considering material surfaces, the concept of the Bidirectional Scattering Dis-
tribution Function (BSDF [13]) is a useful analytical tool for describing the directional
manner in which a material surface scatters electromagnetic (EM) radiation [11]. In this
work, the theory, measurement and modelling of the BSDF, particularly as it relates
to material surface topography has been investigated. This is of special interest since
it describes the eect of undesirable light scattering in optical imaging systems [7]. In
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critical applications such as space telescopes, unwanted stray light reaching the image
sensor reduces the quality, contrast and viability of the image [14]. This applies par-
ticularly to imaging systems operating at short wavelengths such as lens systems for
ultraviolet lithography, a key enabling technology in the manufacture of high density
integrated circuits [15]. Shorter wavelengths are more susceptible to scattering from
small topographic surface features as well as small particles.
When considering high performance optical imaging applications that are sensitive
to stray light, a few questions need to be answered:
1. Is there a practical and reliable way of computing the BSDF of an optical surface
from measurements of the surface topography? It is signicantly important to
answer this question appropriately to successfully implement a precision imaging
system for the following reasons:
(a) Surface topography is comparatively easy to measure since commercial in-
struments for surface prolometry (down to atomic scale) are readily avail-
able.
(b) BSDF is comparatively dicult and slow to measure since it must be meas-
ured over a large range of incidence and scatter angles as well as over dierent
wavelengths.
(c) High precision instruments for direct measurement of the BSDF (scattero-
meters [16]) are typically research instruments designed, implemented and/or
operated by metrological or academic institutes [17].
(d) It is much more practical and cost-eective to specify surface topography
statistics rather than BSDF when optical surfaces are manufactured,
(e) When undertaking computational analysis of the performance of a complete
optical imaging system, it is the surface BSDFs that are required.
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2. Are there any ways in which scatterometers can be improved such that direct
BSDF measurements of optical surfaces or components could become more routine
and accessible? Resolving this question is important because:
(a) Surface topography is not the only factor contributing to unwanted scatter
from optical components. Other contributors include sub-surface damage,
optical coating structure and internal (bulk material) scatter such as from
bubbles and artefacts within glass components, which implies that scatter
produced through these other mechanisms is visible to scatterometers but
not via surface topography measurements.
(b) Direct BSDF measurements are required for validation of the models and
methods used for computing BSDF from surface topography (see preceding
question).
3. What are factors that can degrade the performance of an optical system that
is sensitive to stray light? It is important to know and understand all of the
possible mechanisms by which scattered stray light can originate in precision
optical systems as well as other factors that can aect performance.
4. What are the most advanced current methods for predicting and measuring the
performance of critical, complex imaging systems which are very sensitive to both
optical aberrations (essentially geometric errors) and stray light? This question
is highly relevant to state-of-the-art, short wavelength imaging systems such as
those used in deep and extreme ultraviolet (DUV and EUV) lithography.
The main objective of this work is therefore to explore possible answers to the above
questions.
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1.2 Precision Imaging Optical Systems
Of particular interest in precision optical imaging systems is the BSDF of the lens and
mirror surfaces. Precision imaging optics have a broad range of applications such as
microscopy, imaging astronomy, Earth Observation (EO) from space and lithography
[15, 1820]. These systems normally comprise many optical surfaces and components
and thus the eect of unwanted scatter from these surfaces is cumulative (see Figure
4.1 for an example of a lithography lens system).
Lithography is a particularly demanding imaging technique applied in the man-
ufacture of state-of-the-art electronic Integrated Circuits (IC) and thus the eect of
unwanted scatter must be well-known and accounted for. In fact, in any of the above
applications, undesirable scattered light from the optical surfaces, within the bulk of
the optical components or from mechanical components, could reach the image plane.
Should such scattered or stray light reach the image plane, the quality (contrast, dy-
namic range) and utility of the images produced by these optical systems is signicantly
reduced [7]. With ever increasing demand for better image quality at short wavelengths,
such as in extreme ultraviolet (EUV) lithography [20], it becomes increasingly necessary
to have powerful techniques for measuring weak scatter from single optical surfaces as
well as the cumulative eect of weak scatter in imaging systems.
The instruments used to measure scatter from individual optical surfaces or within
complete optical systems are referred to as scatterometers [11, 16]. When considering
precision optical surfaces, there is an implicit assumption that upon interaction with
the surface, it is possible to dierentiate the departing radiation into scattered and
unscattered components. Scattered radiation must be very weak to maintain high image
contrast. The unscattered component, also referred to as the specular component is
much more intense in this situation and thus results in a clearer image. A scatterometer
is therefore extremely useful in distinguishing between these two components [11] and
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thus must have very high measurement dynamic ranges, in order to discern weak scatter
departing from the surface in directions very close to the direction of the specular
beam [21].
The work presented herein began as an eort to rene the techniques for addressing
the problems in measuring weak scatter near the direction of the specular beam. How-
ever, it soon became clear that the classication of light into scattered and unscattered
components is essentially arbitrary. In reality, there is a continuum of scattering into
wide angles by small surface topographic features or particles through to narrow angle
scatter from the limiting aperture of the optical beam (diraction). Nevertheless, there
is a need to assume that there is a measurable distinction between random or incoherent
scattered light and the specular or coherent component [21]. The random/incoherent
scattered component may also be referred to as diusely scattered light [11].
Due to diraction [22], the limiting aperture of an imaging system, together with
the wavelengths of light utilised by the system determine the limit of ne detail that
can be discerned in the image.
Imaging optical systems can be idealised as linear and shift-invariant systems [23]
for many purposes and can thus be attributed with a transfer function [24, 25], called
the Optical Transfer Function (OTF). The Fourier Transform (FT) of the OTF is the
system Point Spread Function (PSF). The PSF expresses the spatial intensity of the
image of a perfect point of light taken as the object. The PSF and the OTF are an FT
pair that both describe the quality of the image.
The above problem can thus be reduced to a requirement for performing very high
dynamic range measurements of the PSF/OTF of an imaging system.
The following chapters consider the measurement and applications of the BSDF
concept. Thereafter, the measurement and specication of scattering from optical sur-
faces is discussed in the context of performing high dynamic range measurements of the
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PSF. This nally leads to the problem of using such PSF measurements to extract lens
error (aberration) diagnostic information from the precision PSF measurements. The
latter task is referred to as aberration retrieval.
1.2.1 Aberration Retrieval
Traditionally, laser interferometry has been a powerful technique for performing aberra-
tion retrieval [26,27]. Interferometry can be used to measure the shape of the wavefront
emerging from the exit pupil of the imaging system under test. The emergent wave-
front shape is quantied as the pupil function [28]. The pupil function is also related
to the PSF in that the PSF is the Fourier transform of the autocorrelation of the pupil
function (see Figure 2.2).
However, conventional interferometry for evaluation of optical system performance
suers from a number of drawbacks:
1. Usage of a laser as a diraction-limited point source restricts measurement to a
single wavelength. Many optical systems of interest are polychromatic in that
they respond to a range of wavelengths and the PSF/OTF always varies with
wavelength [24].
2. Long coherence length lasers are generally required to obtain decent interferogram
contrast with large path dierences. Such lasers inherently produce noisy images
due to laser speckle phenomenon [27].
3. Interferometers tend to produce spurious fringe patterns from stray (ghost) reec-
tions in the system and from dust particles or defects on the lens/mirror surfaces.
4. The spatial resolution of the camera used to record the interferogram limits the
range of spatial frequencies represented in the measurement [27,29].
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5. The PSF is only indirectly measured by performing an autocorrelation of the
pupil function computationally reconstructed from the interferogram [27].
Spurious fringes and laser speckle increase the noise in interferometric measurements,
reducing the overall signal-to-noise ratio of the measurement result [27] and hence
increasing the uncertainty.
The capability to measure the PSF directly and also perform diagnostic aberration
retrieval is an attractive alternative proposition. Unfortunately, this is inherently an
ill-posed problem because autocorrelation is generally an irreversible operation. The
pupil function carries aberration information and the PSF is the Fourier transform
of the autocorrelation of the pupil function. However, there is a growing body of
knowledge which makes use of intensity measurements in the vicinity of the focal plane
to overcome this barrier. Of special interest in this regard is the Extended Nijboer-
Zernike (ENZ) approach which allows for accurate computation of the EM eld in
the focal plane region. Aberration retrieval is therefore facilitated by leveraging the
measurement diversity made accessible by the ENZ approach.
The journal article, submitted to Current Applied Physics and which is reproduced
in Chapter 6, is the centrepiece of this MSc. It arises inter alia from consideration of an
additional or alternative method for achieving measurement diversity in order to make
use of the ENZ approach for aberration retrieval.
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Chapter 2
Theoretical Background
2.1 Introduction
In this chapter, relevant denitions, theory and models for the BSDF are presented.
The rst question posed in the introduction to Chapter 1 (Is there a practical and
reliable way of computing the BSDF of an optical surface from measurements of the
surface topography?) is explored and further discussed.
In addition to seeking an answer to this question, a generic method is presented for
using such predicted BSDFs in stray light analysis of precision optical imaging systems.
2.2 Denitions and Terminology
Much of the formal terminology for scattering was established by Nicodemus et. al
[13] at the National Institute of Standards and Technology (NIST) in 1977. Details
of the denitions and notation may vary amongst the various disciplines making use
of these or similar concepts. In part, this is due to large dierences in how these
concepts are applied amongst these disciplines and because the literature has developed
independently, to some extent, in each application area.
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2.2.1 Bidirectional Scattering Distribution Function
The Bidirectional Scattering Distribution Function (BSDF) is dened [13] as the dif-
ferential spectral radiance, dLλ departing (scattered) from the surface in a particular
direction (θs, φs) divided by the dierential spectral irradiance, dEλ arriving at the sur-
face in a particular direction (θi, φi). The directions of arrival and departure are dened
using the polar angle relative to the surface normal and the azimuthal angle relative to
a specied azimuthal reference,
BSDF (θi, φi, θs, φs;λ) =dLλ(θs, φs)
dEλ(θi, φi). (2.1)
In general, the BSDF is a function of the directions of both the incident and scattered
light directions (hence bidirectional) as well as the electromagnetic wavelength (λ)
dependence.
The BSDF of a material is not directly measurable, since light sources and detectors
in reality have nite angular apertures or linear extent. Therefore the measured quantity
is better described as the Biconical Scattering Distribution Function (BCSF) [13, 30].
This is a common feature of many kinds of measurements in which a physical quantity
is not measured instantaneously but sampled over a nite spatial or temporal interval,
often in a non-uniform manner [29, 31]. This weighted sampling, performed by the
measuring instrument is quantied by the apodization or instrument function [32]. The
instrument function can be dened as the Fourier Cosine Transform (FCT) of the
apodization function.
Apodization functions are referred to by other names in dierent contexts or discip-
lines. For example, when describing the relative spectral response of an optical detector,
the spectral apodization function may be called a Spectral Response Function (SRF) or
Relative Spectral Response (RSR). In signal processing, the apodization function may
9
be called a window function or tapering function [33].
In this respect, BSDF is not fundamentally dierent to other physical measurands
such as optical power spectra which are thought to have a true underlying value, which
is imperfectly measured [34]. Accepting that the BSDF is not directly measurable,
when referring to BSDF measurements, this should be understood to include any real
measurements that are weighted spatial or spectral averages of the true BSDF.
2.2.2 BRDF and BTDF
The BSDF is the more general form of the bidirectional scattering function. It includes
the possibility of (transmitted) forward scatter into the hemisphere beyond the hemi-
sphere of arrival at the surface, as well as (reected) backscatter into the hemisphere of
arrival. The BSDF can therefore be split into two functions, the Bidirectional Reect-
ance Distribution Function (BRDF) and the Bidirectional Transmittance Distribution
Function (BTDF) [13].
In the literature the formal dierences between BSDF, BTDF and BRDF are not
always apparent and the terms can be used informally or interchangeably. In the eld
of Earth observation from space or airborne platforms [30], the directional reectance
of the Earth surface is commonly and correctly referred to as the BRDF. In computer
graphics [2], BRDF is the most commonly used term, although transparent or translu-
cent objects may occur.
In relation to optical systems, the BTDF is usually the more relevant for lens sur-
faces, while the BRDF is more relevant for mirror surfaces for obvious reasons. When
it comes to the opaque mechanical parts of a lens system, the BRDF (of black paints
for example [35]) is formally correct and almost exclusively the term that is used in
practice.
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Figure 2.1: Scattering Geometry
2.3 Denitions Specic to Optical Surfaces
When dealing with the optical surfaces of lenses and mirrors, there are some specic
geometrical denitions relating to directional scattering that become relevant and that
are commonly used in related literature [11].
Figure 2.1 shows a at, reective optical surface with unit normal vector n pointing
away from the surface. The following denitions largely follow Dittman [36] and the
Zemax® manual [37]. A plane electromagnetic wave travelling in the direction desig-
nated by the unit vector I intercepts the surface at an angle of incidence θi measured
relative to the surface normal. The coherent, specular reection leaves the surface
travelling in the direction given by the unit vector R at an angle of θr relative to the
surface normal vector. The unit vectors n, I and R all lie in a plane called the plane of
incidence [22]. The unit vector S indicates a possible direction of departure for inco-
herent, random optical scatter occurring at the surface. Random scatter may occur due
to residual surface fabrication roughness, a surface defect such as a scratch, or surface
contamination such as a dust particle. An important quantity that will arise often in
11
the discussions to follow is the sine magnitude of the scatter angle. This is a scalar
quantity, designated β, dened as the magnitude of the dierence vector between the
projections of R and S onto the plane of the surface. In Figure 2.1, the projection
vector of R onto the surface plane is labelled ~β0 and the projection of S is labelled ~β.
The sine magnitude of the scatter angle is dened as:
β = |~β − ~β0|. (2.2)
The Zemax® manual [37] denes ~x = ~β − ~β0 and x = |~x|, but this notation will be
avoided and β will be used here and again in 2.5.1. Note that β = 0 for scattering in
the specular direction and that β can approach a maximum value of 2 for the extreme
case of backscattering (S = I) of light at grazing incidence (θi → 90).
The signicance of β is best understood in terms of rst order diraction due to a
grating having a frequency of f repetitions (cycles) per unit distance in the plane of
the surface. For normal incidence (θi = 0), the simple grating equation [22] gives the
relationship between f and the rst order diraction scattering angle θs as follows:
fλ = sin θs, (2.3)
where λ is the wavelength of light and β is the generalisation of sin θs to arbitrary
scattering directions and angles of incidence as dened in Equation 2.2. For a particular
wavelength, every direction of scatter away from the specular direction can be associated
with rst order diraction from a grating on the surface having a particular spatial
frequency and azimuthal orientation. β is the link between the angle of scattering, the
spatial frequency and the wavelength and it retains the simple property that for all
angles of incidence and scattering:
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β = fλ. (2.4)
The above denition also applies to refractive transmission through an optical sur-
face such as the surface of a lens. In general, the surfaces of lenses and mirrors are
curved, so there is an implicit assumption that the BSDF can be specied and meas-
ured over an area that is suciently small to be treated as locally at.
In many cases of interest to stray light analysis, the surface BSDF can be expressed
as a function of β.
2.3.1 Total Integrated Scatter
The Total Integrated Scatter (TIS) is dened as that proportion of the total radiant
power remaining after light interaction with the surface that is scattered out of the
specular beam [38]. That is, a certain quantity of radiant power (ux) is incident on
the surface, of which some is absorbed and the remainder is reected or transmitted.
The TIS is the ratio of the surviving ux scattered out of the specular beam to the total
surviving (non-absorbed) ux. The most elementary situation is the case of an optical
mirror, where only the reected ux is of interest. Reectance in general is dened as
a ratio of reected to incident ux [13]. If the total reectance for a particular angle of
incidence θi is ρt, the specular reectance is ρr and the diusely scattered reectance is
ρs then the TIS can be expressed as [38]:
TIS =ρsρt
=ρs
ρs + ρr=ρt − ρrρt
= 1− ρrρt. (2.5)
In the context of isotropic mirror surfaces (or if there is interest in reectance only),
the TIS for a particular angle of incidence is also an integral of the incoherent BRDF
as:
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TIS =
¨
BRDF (θi, φi, θr, φr), (2.6)
or as a function of β as:
TIS =2π
ρt
ˆ
BRDF (β)βdβ. (2.7)
It is very important to note that the BRDF used here does not include the spec-
ular (unscattered) beam. In this sense, it does not conform to the denition of
BSDF/BRDF given above in Equation 2.1. An obvious problem with the denition
of TIS here is that, as pointed out above, the distinction between scattered and un-
scattered is rather arbitrary and requires, in addition, some statement about the angle
of scatter that is taken as the boundary between scattered and specular. In the case
of the simple Harvey scatter model discussed in 2.5.1, the boundary angle is taken as
β = 0.01 (see Equation 2.2 for a denition of β), which for modest incident angles is
close to an angle of 0.01 radians from the specular beam. Available literature does not
justify dening this particular boundary and it thus has the appearance of being an
arbitrary choice.
Nevertheless, TIS is still commonly used and measured [38, 39], and this particular
problem is sometimes referred to as bandwidth-limiting of the roughness spatial fre-
quencies [40] or the relevant spatial frequencies. Upper and lower spatial frequency
limits are both relevant. In the case of scatterometer type measuring instruments [16],
the boundary angle is usually given as a specication. TIS can be very sensitive to the
boundary angle and in certain cases this may bring the usefulness of TIS measurements
into question unless the boundary angles are known and reported [38].
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2.3.2 Surface Topography and Power Spectral Density
The strong relationship between TIS and surface roughness has been observed and in-
vestigated in great detail [11]. Optical scatter in general and TIS in particular has
served as a useful means of measuring or characterising surface roughness and topo-
graphy [38,39,4143]. The statistical nature of the surface topography is captured using
the autocorrelation of the topographic function [11]. The autocorrelation function is
the surface topography Power Spectral Density (PSD), which is a function of spatial
frequency, f , along the surface [11]. The denition of the 2-dimensional PSD is [44]
the square modulus of the Fourier transform of the surface topography function z(x, y),
PSD2D(fx, fy) = limL→∞
1
L2
∣∣∣∣∣∣
L
0
L
0
z(x, y) exp[−2π(fxx+ fyy)]
∣∣∣∣∣∣
2
, (2.8)
where fx and fy are spatial frequencies in the x and y directions along the surface.
The dimension L is introduced as the length scale over which the topography has been
measured. The one-dimensional PSD applies to surfaces of isotropic characteristics
calculated by averaging the 2-dimensional PSD over all azimuthal directions along the
surface as:
PSD(f) =1
2π
2πˆ
0
PSD(f, Ψ)dΨ, (2.9)
where f =√f 2x + f 2
y and Ψ is the azimuthal angle along the surface i.e. Ψ = arctan(fy/fx).
The eective surface roughness for normal incidence and isotropic roughness [38] is:
σeff =
2π
fmaxˆ
fmin
PSD(f)fdf
12
, (2.10)
where the integration limits represent the relevant spatial frequencies in any particular
situation. In the limit as fmin → 0 and fmax → ∞, the eective roughness converges
15
to the total roughness σ. The TIS can be calculated from the eective (or relevant)
roughness using the classic relationship [38],
TIS = 1− exp
[−(
4π cos θiσeffλ
)2], (2.11)
which for smooth surfaces can be approximated as:
TIS ≈(
4π cos θiσeffλ
)2
. (2.12)
2.3.3 Autocovariance Function
An alternative but equivalent means of describing surface topography besides the PSD
is the surface topography autocovariance function (ACV). For surfaces of zero mean
height this is equivalent to the surface topography autocorrelation function. The surface
ACV and PSD are a Fourier transform pair whereby ACV is used for the purpose of
quantifying the surface characteristic correlation length τc. This is the lateral spacing at
which the ACV falls to 1/e of the maximum value [44]. It is also used as a computational
stepping stone to the surface PSD as illustrated in Figure 2.3.
2.3.4 Angular Resolved Scattering
A variant of the BSDF for describing the angular scattering of a surface is the Angular
Resolved Scattering (ARS, [38,44]), which arises in the literature. The ARS is dened
as the optical ux component ∆Φs scattered into a solid angle ∆Ωs in the scattering
direction θs normalised to the solid angle and incident ux Φi i.e.
ARS(θs) =∆Φs(θs)
∆ΩsΦi= BSDF (θs) cos θs = BSDF (θs)γs. (2.13)
16
2.3.5 Surface PSD Relationship to Image Quality
The relationship between surface topography and image quality is one of special signi-
cance to the performance of precision imaging systems.
Precision imaging systems can be conceptualised as a sequence of surfaces (inter-
faces) between bulk materials which manipulate or modify the electromagnetic wave-
front as it traverses the surfaces in turn. For example, the topographical features of
each surface are cumulatively imprinted on the wavefront by the process of optical phase
delay.
Topographical features on the optical surfaces which are much smaller than the
wavelength of light down to atomic scale have little eect, causing low-level back-
ground Rayleigh scatter which is fairly isotropic [45]. For larger, more relevant spatial
frequencies of such topographic features, the PSD of the surface topography is a use-
ful characterisation as illustrated above. The shape of the wavefront (dened by the
complex exit pupil function, P ) emerging from the imaging optical system, which sub-
sequently converges to the focal plane, carries the signature of the PSDs of all the
surfaces through which it has passed. It is instructive to note that the system Optical
Transfer Function (OTF) is the autocorrelation of the pupil function [24].
The optical system PSF is the Fourier transform of the OTF and it is the PSF/OTF
Fourier transform pair which encodes the quality of the image. The PSD of the optical
surfaces at all relevant spatial frequencies is hence directly linked to the image quality.
Figure 2.2 shows the relationships between the complex exit pupil function P , the Amp-
litude Point Spread Function (APSF) which is the complex amplitude of the PSF, the
Modulation Transfer Function (MTF), OTF and Phase Transfer Function (PTF). The
symbol F denotes a Fourier transform relationship. The symbol ? denotes correlation.
In optical systems such as those for deep or extreme UV lithography [15,19,20], the
wavelength of light is very small and the relevant bandwidth of the surface PSD is relat-
17
MTF
P
F
P?P // OTF
|OTF |
OO
F
arg(OTF ) // PTF
APSF
OO
|APSF |2 // PSF
OO
Figure 2.2: Optical Transfer Function Relationships
ively wide, spanning many orders of magnitude [46]. A number of dierent measurement
techniques are required to cover such a broad spatial spectrum, usually incorporating
conventional interferometry [26], surface-proling white-light interferometry [47] and
atomic force microscopy (AFM [48]).
Within the realm of precision optical systems, the relationship between surface
PSD and BSDF is of particular importance. This is simply because there are many
commercial proling instruments for measuring surface topography down to atomic
scale. On the other hand, instruments for precision high dynamic range measurements
of surface BSDF are rare and are usually special systems designed and operated by
metrological or academic institutes [17,4952].
2.4 Optical Scatter
Several theoretical approaches to scatter from moderately rough to nearly smooth sur-
faces have been published [44, 53]. These approaches are typically developed from
Maxwell's equations using various simplications and approximations [11]. While not
18
completely rigorous, these theories are very useful both for gaining insight into the
scattering phenomenon and for practical purposes. In some cases they also provide a
means for solving the inverse problem, namely that of inferring the statistical proper-
ties of the surface prole through goniometric or spectro-goniometric scatterometry [39].
Of note are theories known as the classical Beckman-Kirchho (BK) theory [54], the
Rayleigh-Rice (RR) theory [55], the Harvey-Shack (HS) theory [56] and the most re-
cent Generalised Harvey-Shack (GHS) theory [57, 58]. The RR and GHS theories will
be presented briey to illustrate the relationship between the Power Spectral Density
(PSD) of the surface topography and the BSDF.
2.4.1 Rayleigh-Rice Scattering
Also known as the vector perturbation approach, the Rayleigh-Rice theory [11, 55] ap-
proaches rigour in the smooth surface limit. It provides a very direct relationship
between surface prole and angular scattering and is the cornerstone of statistical
smooth surface proling using goniometric scatterometry. The relationship is as fol-
lows:
BSDF (β) =4π2∆n2
λ4QγiγsPSD(f). (2.14)
The BSDF in this case is a function of the (rst order diraction grating equivalent)
amount, β, by which the ray is scattered from the specular direction by a surface spatial
frequency component of f cycles per unit distance. β and f are related through the
wavelength simply as β = fλ as explained in 2.3. The change of refractive index
between ray arrival and ray departure from the surface is ∆n. For reecting surfaces,
∆n = 2.
Q is a reectance factor that generally depends on the optical constants of the scat-
19
tering surface substrate, the scattering geometry and the polarisation state. For small
incident and scattering angles, this factor reduces to just the total surface reectance
ρt.
The PSD is expressed here as a function of spatial frequency in any direction along
the surface which is only possible for isotropic surfaces having no azimuthal dependence
of scattering. This is the radial prole of the 2-dimensional PSD. Notice that the
BSDF drops o with the cosines of both the polar angles of incidence (γi = cos θi)
and scattering (γs = cos θs). Note also that since the PSD of a zero mean height
distribution typically levels o at a spatial frequency of f = 0, the BSDF also levels o
at the specular direction where β = 0. Clearly, this BSDF model excludes the specular
component.
2.4.2 Generalised Harvey-Shack Scattering
Recently introduced by Krywonos, Harvey and Choi [57, 59], the Generalised Harvey-
Shack (GHS) theory is a scalar treatment using the Helmholtz equation [28] and lin-
ear systems theory to describe scattering from smooth through to moderately rough
surfaces. The theory is valid for arbitrary forms of the PSD (but with Gaussian to-
pographic height statistical distribution) and large angles of incidence and scattering.
They introduced a surface scattering transfer function [57,59]:
Hs(x, y, γi, γs) = exp
− [2πσeff (γi + γs)]
2
[1− ACV (x, y)
σ2
], (2.15)
where x, y and σeff are wavelength normalised coordinates and eective surface rough-
ness. The ARS then takes the form of a correctly scaled Fourier transform of the surface
20
z(x, y) z?z // ACV
1/e Point // τc
GHSBSDF PSDGHSTheoryoo
F
OO
RRTheory // RRBSDF
Figure 2.3: Surface Function Relationships
scattering transfer function as:
ARS(θs) = QγsF Hs(x, y, γi, γs) . (2.16)
The reectance factor Q is dened in the same manner as that used in the RR
relationship in Equation 2.14. F is again the Fourier transform operator.
The GHS approach combines the advantages of the RR and BK approaches while
avoiding the limitations of either of these theories. In addition, the theory provides a
practical, if computationally expensive means of translating surface topographic prole
measurements into an ARS/BSDF model for analysis of stray light eects in imaging
systems.
The general relationship between the surface topography function z(x, y) , the sur-
face ACV , the surface PSD and the surface BSDF is illustrated in Figure 2.3. The
symbol ? denotes the correlation operation.
2.5 Scatter Models for Stray Light Analysis
There are a number of direct models available for scatter resulting from residual fabrica-
tion roughness on otherwise smooth optical surfaces. The models that will be discussed
below are the 2-parameter and 3-parameter Harvey models [60] and the 3-parameter
K-correlation model [11,36,61].
21
2.5.1 The Harvey Models
The 2-parameter Harvey model [60] is a simple inverse power law relating the scatter
angle sine magnitude β to the BSDF as follows:
BSDF (β; b, s) = b(100β)−s. (2.17)
The rst parameter, b is the value of the BSDF at a scatter sine angle magnitude
of β = 0.01, which is near enough to a scatter angle of 0.01 rad measured from the
specular direction. For optical surfaces, typical values of b are 0.01 < b < 1. The
second parameter, s is the log-slope of the BSDF decline with respect to β which for
optical surfaces lies typically in the range of 1 < s < 3. This simple model appears to
be fractal in the sense that the RR scatter theory predicts a surface radial PSD (see
Equation 2.14) that is also a simple inverse power law with the same log-slope across
all spatial frequencies. This model is problematic near the specular direction, as it
increases without limit as β → 0. This behaviour is open to interpretation. It could
mean that the RR PSD also increases without limit as f → 0, or that the geometrical
specular part of the BSDF is included in the model. The rst interpretation is not
consistent with smooth optical surfaces and the second interpretation is not useful. For
this and other reasons, the 2-parameter Harvey model presents a consistency problem
for the denition of the TIS. Although it has proven very useful in stray light analysis,
is should be avoided in favour of the 3-parameter Harvey model or the K-correlation
model.
For real optical surfaces, the incoherent BSDF levels o at the specular direction
and the 3-parameter Harvey model provides this behaviour,
BSDF (β; b0, l, s) = b0
[1 +
(β
l
)2]−s/2
. (2.18)
22
This 3-parameter model levels o to a value of b0 in the specular direction as β → 0.
At the shoulder value of β = l, the BSDF starts tending towards the simple inverse
power law behaviour of the 2-parameter model, also exhibiting the log-log slope of s.
2.5.2 The K-Correlation Model
As one of the available optical surface scatter models in the Zemax® optical design
and analysis code [37,62], the K-correlation model warrants special attention. Used by
Church [61, 63] and Stover [11] to describe scatter from optical surfaces, the practical
use of the model (also known as the ABC model) has been facilitated by Dittman [36]
and Gangadhara [64] for the purposes of stray light modelling in imaging systems. The
K-correlation 1-dimensional PSD is parametrised [36] as,
PSD1D(f) =A
[1 +B2f 2]s−1
2
, (2.19)
and the prole of the 2-dimensional PSD as,
PSD2D(f) =ABg
[1 +B2f ]s2
, (2.20)
where,
g =Γ(s/2)
2√πΓ( s−1
2). (2.21)
Γ(x) is the Gamma function [65] and s is the mid-frequency log-slope of the PSD as for
the Harvey models. The parameter B is related to the characteristic surface wavelength
(correlation distance τc) of the surface irregularities [38] and A is a normalisation factor.
The K-correlation model provides that the PSD and hence also the BSDF follow an
inverse power law over the mid-section of the spatial spectrum, described as fractal
and commonly the result of subjecting glassy materials to conventional optical (grind
23
and polish) manufacturing methods [61,66].
Dittman [36] provides explicit expressions for A by normalising the PSD to the
total or eective surface roughness, σ using the RR relationship to BSDF and from
there proceeds to analytical expressions for the BSDF (not reproduced here).
A convenient feature of the K-correlation model is that the ACV (the Fourier trans-
form of the PSD) which appears in the GHS expression (Equation 2.15) can be calcu-
lated analytically from the K-correlation t parameters (A, B and s) using [67]:
ACV (r) =√
2πA
B
2(1−s)/2
Γ((s−1)/2)
(2πr
B
)(s−2)/2
K(s−2)/2
(2πr
B
), (2.22)
where Kn(x) is the modied Bessel function of the second kind [65] and order n.
2.5.3 Scatter Model Selection
The choice of scatter model depends on the accuracy with which the scatter must be
accounted for in the application. For critical analysis of stray light eects in imaging
systems, the K-correlation model is preferred. However, model selection also depends on
the data type that is available for compilation/analysis of the model. The 2-parameter
Harvey model provides unrealistic behaviour near the specular direction and should
be avoided in favour of the 3-parameter Harvey model if the small-angle roll-o of the
BSDF is known or can be determined. Likewise, if information about the large angle
roll-o is also available, then the K-correlation model should be used in favour of the 3-
parameter Harvey model. Selection of model can be assisted by checking the sensitivity
of analysis results to the input K-correlation parameters.
24
2.6 Surface Prolometry and BSDF
There is a strong need in the optical modelling community, especially in precision ima-
ging systems, to have the capability to synthesise a reliable surface BSDF model from
surface prole (topography) measurements [61,68]. Such a BSDF model allows for reli-
able stray light analysis of precision imaging optical systems without having to perform
direct measurements of the surface BSDF [69]. Recent developments have provided an
elegant process for moving from surface topography measurements to surface BSDF. It
involves the combination of the Generalised Harvey-Shack scattering theory (see 2.4.2)
and the K-correlation scatter model (see 2.5.2).
The general process in this respect is as follows:
1. Measure the surface topography z(x, y) and compute the PSD over the relevant
spatial frequencies using Equation 2.8. For optical systems operating at short
wavelengths, a combination of measurement techniques may be required to cover
all relevant spatial frequencies as in [46]. These techniques include conventional
surface interferometry [26], white-light surface proling interferometry [47] and
atomic force microscopy [48].
2. Fit a parametrised K-correlation model to the measured PSD. The K-correlation
model is described in more detail in 2.5.2. The GHS theory of scattering is a
linear systems theory and this implies (by denition) that a linear combination
of K-correlation PSDs can be tted if a single t is inadequate.
3. Optical design software such as Zemax® may allow for direct input of the K-
correlation parameters for surfaces in the lens model [64]. However, in the case
of Zemax® [37], this currently means that the RR formulation for the BSDF
(Equation 2.14) will be used [36]. That would also imply that only a single K-
correlation model can be applied. This is generally adequate for well-polished
25
surfaces at moderate-long wavelengths.
4. Calculate the ACV analytically from the tted K-correlation model(s) using Equa-
tion 2.22.
5. Compute the surface transfer function Hs for all relevant angles of incidence and
scattering using Equation 2.15. Good estimates of eective and total surface
roughness would be available from the surface topography/PSD measurements.
6. Calculate the ARS using Equation 2.16. If the BSDF is required, use the relation
given in Equation 2.13. If the PSD spans a very large range of spatial frequencies,
such as in EUV applications, the Fourier transforms used in the process have to be
calculated in log space. The FFTLog algorithm can be used for this purpose [67].
7. The compiled ARS or BSDF can be used in optical simulations to determine
the irradiance distribution in the image plane of an optical system. Of special
interest in this regard is the recent work of Choi and Harvey [46,70] in which they
demonstrate computation of the imaging system PSF in analytic form in terms of
convolutions of the geometrical PSF and the scaled BSDFs of the optical surfaces.
The above procedures provide a means of moving from surface prolometry (topography
measurements) to a very general BSDF model that is valid for a wide range of incident
and scattering angles and for smooth to moderately rough surfaces. These BSDF models
can be used to compute the imaging system PSF by analytical means [46] or through
raytracing with software such as Zemax®. The process described above is illustrated
graphically in Figure 2.4.
The inverse problem of surface quality specication [71] is also very important.
That is, given a system stray light performance metric, what roughness is allowable on
the optical surfaces? However, if there is a suitable solution to a forward problem as
26
described in the above procedure, then solution of the inverse problem often becomes
tractable merely through repeated application of the forward solution.
27
Surface Interferometry
Proling Interferometry
ttz(x, y)
z?z
Atomic Force Microscopyoo
Measured ACV
F
Measured PSD
Fit
K-Correlation PSD
Equation 2.22 A,B,s
A,B,s // Zemaxr // RR PSF
Fitted ACV
Equation 2.15
Hs
Equation 2.16
Optical Layout
OO
ARS
Equation 2.13
BSDFTabulated //
Zemaxr // GHS PSF
Choi and Harvey [46,70]
22
Figure 2.4: Surface Prolometry for Image Quality Modelling
28
Chapter 3
Measurement of Weak Scattering in
the Specular Beam
3.1 Introduction
Measurement of incoherently scattered radiation very close to the coherent specular
beam is challenging due to interference from the specular radiation itself [11]. Rene-
ment of measurement techniques and scatterometer design combined with very careful
geometrical denition of the source beam and sensor apertures have allowed measure-
ments to be performed progressively closer to the specular direction [11,21].
Many scatterometers are congured to accept a at (plano) optical sample having
a surface from which some scatter is expected [16]. The instrument signature is any
signal generated by the scatterometer which is not attributable to scattering at the
sample. The instrument signature is usually measured before insertion of the sample
and subtracted from a measurement performed with the sample in place. The sample
is assumed to be optically inactive except for some small incoherent scatter. This is
never completely true as the sample will also introduce some fresnel reections [22]
29
and optical aberrations that alter the intensity and spatial distribution of the emerging
specular beam, possibly also introducing ghost beams [16].
The requirement for a plano sample is a rather limiting one, thus implying that
lens and mirror surfaces with optical power cannot be directly evaluated for scatter.
Instead, a surrogate, plano sample is used. This surrogate sample should be fabricated
using the same process as the curved lens or mirror surfaces. Some uncertainty is
therefore introduced into the process because the component used in the system is not
the component that was evaluated for scattering.
Ultimately there will be an interest in the overall scattering performance of the
complete optical system, possibly comprising many curved surfaces on a number of
dierent optical substrate materials and with dierent thin lm coatings. If the scat-
tering characteristics of the individual surfaces are known, it is possible to model the
overall performance of the system, but as noted above, there is substantial room for
uncertainty. Therefore it is desirable to be in a position to measure the overall in-eld
stray light performance and to have a powerful means of discriminating undesirable
scattered radiation from the primary specular signal. The instrument for measurement
of this scattered radiation in a complete optical system will be referred to as a system
scatterometer, as opposed to a surface scatterometer intended to evaluate single plano
optical samples (the usual meaning of the term scatterometer [16]).
In this chapter, the possibility of using segmented pupil, focal plane interferometry
will be explored as a means of better discriminating the in-eld scattered stray light
in imaging systems. This is based on the premise that there is a measurable dier-
ence between unscattered (specular, coherent) and randomly (incoherently, diusely)
scattered light. In the following development, this will be taken to mean that the
specular beam exhibits interference eects while the diusely scattered light does not.
30
3.2 Segmented Aperture Interferometry
There are a number of optical instruments and applications in which the pupil or aper-
ture of an optical system is split into several segments (spatial regions) with dierent
phase shifts applied to each segment [72]. Interference eects are then observed in the
image plane of the system. An example of this technique is the Rayleigh interfero-
meter [28]. It is commonly used to make precise measurements of the refractive indices
of gases. More recently, particularly interesting examples are the dual or multiple
aperture space telescope system concepts congured as Bracewell interferometers [73]
for the detection of Earth-like exoplanets. This type of interferometer uses a pair of
apertures on a wide baseline (20 m up to 200 m) to produce a deep interference null
of extremely high viewing resolution within the image. The light from the host star
is thus suppressed while light from a nearby planet can be detected and analysed. It
is this same principle that is applied here to suppress the specular beam in order to
measure the nearby weak incoherent scatter.
A simple type of segmented pupil, phase-shifting interferometer is one that splits the
beam of an optical system in two and introduces a variable amount of phase retardation
(wavefront piston) into one half of the beam. An optical phase retarder for this can
be constructed by diamond-sawing a high quality plano-parallel glass plate in half,
mounting the two pieces side-by-side in the optical beam and then rotating one half
about an axis perpendicular to the cut (or about any convenient, local axis).
A simple, segmented aperture phase retarder of this type is illustrated in Figure
3.1. Rotation of one half of the retarder will alter the phase shift relative to the other
half. The actual axis of rotation makes little dierence, since it is the greater eective
optical thickness of the rotated half that causes the phase shift.
Suppose a uniformly illuminated, plane-wave, monochromatic beam of light passes
through the retarder (Figure 3.1). Assume that the slight lateral displacement of the
31
Figure 3.1: Segmented Aperture Phase Retarder
beam caused by the plate rotation is negligible. The last assumption can be realised
if the system aperture stop lies beyond the retarder and the retarder is adequately
overlled by the incident beam. The phase delay in the EM eld caused by the tilted
plate can be written as a complex factor e−i∆φ, where ∆φ is the phase delay expressed
in radians.
The phase delay can also be written in terms of the change in optical path length
through the tilted plate as a function of the tilt angle, θx as:
∆φ =2πnt
λ
(n√
n2 − sin2 θx− 1
). (3.1)
The thickness of the plate is t and the refractive index referenced to the surrounding
medium is n. A more detailed derivation of Equation 3.1 is provided in A.1.
The far-eld EM amplitude for an aperture with this phase modulation can be
computed using the Fraunhofer diraction integral. The Fraunhofer approximation can
be expressed as [28]:
32
Figure 3.2: Coordinate System
U(x2, y2) =eik∆zei
k2∆z
(x22+y2
2)
iλ∆z
∞
−∞
∞
−∞
U(x1, y1)e−ik
∆z(x1x2+y1y2)dx1dy1, (3.2)
where U is the scalar, time-independent eld amplitude in the source plane when ex-
pressed as a function of (x1, y1) being the coordinates in the source plane and the scalar
eld amplitude in the (far-eld) observation plane when expressed as a function of ob-
servation plane coordinates (x2, y2). The coordinate system is illustrated in Figure 3.2.
It is further assumed that optical propagation occurs chiey in the z-direction, that
the propagation distance between the source and observation planes is ∆z, the optical
wavelength is λ and the angular wavenumber is k = 2π/λ.
The scalar eld amplitude in the source plane U(x1, y1), (taken to be directly after
the phase retarder) can be written as the sum of two vertical apertures (strips running
in the y-direction) multiplied by a single horizontal aperture (running along the x-
direction) function as:
U(x1, y1) =
[rect
(x1 − x0
Dx
)+ rect
(x1 + x0
Dx
)e−i∆φ
]rect
(y1
Dy
), (3.3)
33
x1
y1
x0
x0
Dx Dx
DyDy e−i∆φ
Figure 3.3: Dual Rectangular Aperture Pupil Mask
where the two vertical apertures both of width Dx are displaced by an amount x0 on
either side of the optical axis and the total vertical (y) height is Dy.
The rect function is dened as [74]:
rect(x) ≡
1 |x| < 12
12|x| = 1
2
0 |x| > 12
. (3.4)
The arrangement of the two rectangular apertures (pupil segments) is shown in
Figure 3.3. The constraint that x0 = Dx/2 is imposed so that the two apertures do not
overlap.
Setting
u2 = eik∆zeik
2∆z(x2
2+y22), (3.5)
the resulting Fraunhofer integral is separable as:
34
U(x2, y2) =u2
iλ∆z
∞
−∞
∞
−∞
U(x1, y1)e−ik
∆z(x1x2+y1y2)dx1dy1
=u2
iλ∆z
∞
−∞
[rect
(x1 − x0
Dx
)+ rect
(x1 + x0
Dx
)e−i∆φ
]e−i
k∆zx1x2dx1
×∞
−∞
rect
(y1
Dy
)e−i
k∆zy1y2dy1
=u2
iλ∆z
−x0+Dx/2ˆ
−x0−Dx/2
e−ik
∆zx1x2dx1 + e−i∆φ
x0+Dx/2ˆ
x0−Dx/2
e−ik
∆zx1x2dx1
×Dy/2ˆ
−Dy/2
e−ik
∆zy1y2dy1
=u2
iλ∆zDxDysinc
(Dxx2
λ∆z
)(e−
2πiλ∆z
x0x2 + e2πiλ∆z
x0x2−i∆φ)
× sinc
(Dyy2
λ∆z
). (3.6)
3.2.1 Focal Plane Irradiance
If the output beam from the phase retarder is focused with a perfect lens, the paraxial
focal plane amplitude can be computed using the Fresnel approximation [24]. The
Fresnel integral has a complex exponential quadratic term inside the integral in addition
to the linear terms in the Fraunhofer integral (Equation 3.2) as:
U(x2, y2) =u2
iλ∆z
∞
−∞
∞
−∞
U(x1, y1)e−ik
∆z(x1x2+y1y2)ei
k2∆z
(x21+y2
1)dx1dy1. (3.7)
However, in the paraxial approximation of a thin lens of focal length fl, the phase
change introduced in the pupil plane of the lens can also be expressed as a quadratic
term of the pupil coordinates,
35
φl(x1, y1) = − k
2fl(x2
1 + y21). (3.8)
Thus the amplitude distribution in the source plane directly after the lens becomes:
U(x1, y1) = A(x1, y1)V (x1, y1)eφl(x1,y1)
= A(x1, y1)V (x1, y1)e− k
2fl(x2
1+y21),
where A(x1, y1) is the incident amplitude and V (x1, y1) is a real-valued vignetting func-
tion accounting for transmission losses of the lens. The vignetting function is introduced
to cater for situations where the lens does not have equal transmittance over the whole
aperture (i.e. the lens has some spatial apodization). The propagation distance to the
focal plane is ∆z = fl and the Fresnel integral becomes:
U(x2, y2) =u2
iλfl
∞
−∞
∞
−∞
U(x1, y1)e−ik
∆z(x1x2+y1y2)ei
k2∆z
(x21+y2
1)dx1dy1
=u2
iλfl
∞
−∞
∞
−∞
A(x1, y1)V (x1, y1)e−i 2π
λfl(x1x2+y1y2)
dx1dy1. (3.9)
The integral in Equation 3.9 can be interpreted as a Fourier transform F with spatial
frequencies of fx = x2
λfland fy = y2
λfl. If the phase-retarder and dual aperture mask are
placed a distance d before the lens, the phase factor outside the integral, u2 becomes
u2(x2, y2) = ei πλfl
(1− d
fl
)(x2
2+y22). (3.10)
However, the vignetting factor, V , must be transformed to account for the pupil
shift [75] as:
36
zPlaneEM
Wave
PhaseRetarder
y1
Aperture
Mask
Perfect
Lens
y2
Focal
Plane
d = fl ∆z = fl
Figure 3.4: Simplied Telecentric Imaging Layout
U(x2, y2) =u2
iλflF[A(x1, y1)V (x1 +
d
flx2, y1 +
d
fly2)
].
When the phase-retarder and aperture mask are placed at the front focal point of
the lens (called the telecentric stop position), d = fl and the phase factor u2 = 1. The
focal plane amplitude reduces to:
U(x2, y2) =1
iλflF [A(x1, y1)V (x1 + x2, y1 + y2)] . (3.11)
In the cases of interest here, the lens will have a very small vignetting (wavefront
amplitude spatial apodization) eect. The lens vignetting eect could be non-zero,
for example, because not all rays will strike the lens surface at the same angles of
incidence, and the fresnel reection from glass is dependent on this angle [28]. Assuming
the transmission/vignetting factor, V , of the lens can be neglected, and the incident
amplitude, A, is set to the plane wave with segmented aperture phase retardation
function given in Equation 3.3, the resulting image plane amplitude distribution follows
the pattern of Equation 3.6. This simple, telecentric optical arrangement is illustrated
in Figure 3.4.
Hence, under the assumed conditions of monochromatic incident plane wave, uni-
37
Figure 3.5: logE(x2, y2) for ∆φ = π
formly illuminated pupil as well as telecentric, paraxial and unvignetted imaging, the
focal plane irradiance, E(x2,y2), is the square modulus of the focal plane amplitude [28],
U(x2, y2) calculated according to Equation 3.11. That is:
E(x2, y2) =
∣∣∣∣DxDy
iλflsinc
(Dxx2
λfl
)(e− 2πiλfl
x0x2 + e2πiλfl
x0x2−i∆φ)
sinc
(Dyy2
λfl
) ∣∣∣∣2
=
[DxDy
λflsinc
(Dxx2
λfl
)sinc
(Dyy2
λfl
)]2
×
2
[cos
(4πx0x2
λfl−∆φ
)+ 1
]. (3.12)
In the specic case of x2 = y2 = 0 and ∆φ = π, the axial irradiance is zero. That is,
with the retarder set to one half wave, the axial irradiance drops to zero. An example
image of the logarithm of E(x2, y2) for ∆φ = π is shown in Figure 3.5. The linear values
are plotted in green.
38
0.0
0.2
0.4
0.6
0.8
1.0
EE
max
−1 0 1
φl =4πx0x2
λfl
∆φ = 0
∆φ = π
Sum
Figure 3.6: x-direction Cross Section of E(x2, y2)
A cross-section of the normalised irradiance in the x-direction through the origin is
shown in Figure 3.6, where φl = 4πx0x2
λfl. As might be expected, rotation of the phase
retarder will give rise to a travelling interference fringe pattern. The fringes travel in
the direction perpendicular to the cut through the retarder.
3.3 Source and Detector Aperture Eects
The fact that the light source and focal plane detector, in practice, will be of nite
rather than innitesimal size will have an eect on the depth of the irradiance minimum
(interference null) that can be achieved. This in turn will have an important eect on
the system capability to measure low levels of scattered irradiance in the interference
null. In this section, these eects are explored, together with possible methods of
analysis.
39
The result expressed in Equation 3.12 is the irradiance pattern for a uniform, mono-
chromatic plane wave incident on the phase retarder. This is equivalent to a point source
at innity with E(x2, y2) regarded as the system Point Spread Function (PSF). For a
spatially extended source that is spatially incoherent, the linear systems approach [24]
is to convolve the spatial intensity distribution of the source projected to the image
plane with the system PSF. Since the intention here is to exploit the irradiance min-
ima in the system PSF cross-section in the x-direction for the purpose of measuring
scattered light, the source should compromise these minima as little as possible and
should therefore have minimal spatial extent in the x-direction. A slit source with the
long axis in the y-direction is therefore chosen for analysis, since this will have least
impact on the depth of the irradiance null. The dimensions of the slit source, projected
to the image plane will be labelled sx and sy. The convolution (denoted ⊗) of the
system PSF and the projected slit aperture function, S(x2, y2), is thus
Es(x2, y2) = E(x2, y2)⊗ S(x2, y2)
= E(x2, y2)⊗[rect
(x2
sx
)rect
(y2
sy
)].
Here it is assumed that the axes of the source slit are precisely aligned to the axes
of the phase retarder rotation axis and mask apertures (see Figures 3.3, 3.1 and 3.4).
The convolution operation is dened [32] as,
f(x, y)⊗ g(x, y) ≡∞
−∞
∞
−∞
f(x′, y′)g(x− x′, y − y′)dx′dy′. (3.13)
In the case of separable functions the convolution can also be separated as:
40
[f1(x)f2(y)]⊗ [g1(x)g2(y)] =
∞
−∞
∞
−∞
f1(x′)f2(y′)g1(x− x′)g2(y − y′)dx′dy′
=
∞
−∞
f1(x′)g1(x− x′)dx′∞
−∞
f2(y)g2(y − y′)dy′
= [f1(x)⊗ g1(x)] [f2(y)⊗ g2(y)] .
Likewise, if the detector has a rectangular aperture of dimensions px × py with
spatially uniform response, there will be a further convolution with the detector/pixel
aperture function, Pd(x2, y2) = PX(x2)PY (y2), which is separable for rectangular de-
tectors. The resulting function with both detector and source convolutions can be used
to determine the amount of ux falling on the detector aperture, wherever the detector
is positioned in the image. The separated functions are dened as,
Esp(x2, y2) = E(x2, y2)⊗ S(x2, y2)⊗ Pd(x2, y2)
= 2
(DxDy
λfl
)2
[Ex(x2)⊗ Sx(x2)⊗ Px(x2)]
× [Ey(y2)⊗ Sy(y2)⊗ Py(y2)]
= 2
(DxDy
λfl
)2
Exsp(x2)Eysp(y2),
where,
Exsp(x2) = Ex(x2)⊗ Sx(x2)⊗ Px(x2)
=
sinc2
(Dxx2
λfl
)[cos
(4πx0x2
λfl−∆φ
)+ 1
]
⊗ rect
(x2
sx
)⊗ rect
(x2
px
)
41
and
Eysp(y2) = Ey(y2)⊗ Sy(y2)⊗ Py(y2)
= sinc2
(Dyy2
λfl
)⊗ rect
(y2
sy
)⊗ rect
(y2
py
).
These convolutions though dicult to solve analytically1, can be computed numer-
ically using dense sampling of x2 and y2 and without using Fourier methods. This result
could then be compared to a numerical result computed using Fourier methods, or with
Zemax® [37] which uses Fourier methods amongst others.
3.4 Bilaterally Symmetric Segmented Pupils
In this section a more general approach to analysis of segmented pupil interferometry
will be explored, starting with the result that the incoherent Optical Transfer Function
(OTF) can be computed as the normalised autocorrelation of the pupil function [24,25].
The generalised pupil function gives the amplitude and phase of the EM eld at the
exit pupil. In this case the pupil function is taken to comprise two non-overlapping
segments (regions) one of which is a reection of the other about the y-axis. Moreover,
the pupil function is assumed to be symmetrical about the x-axis. The aperture mask
shown in Figure 3.3 is a special case. The pupil function therefore comprises a real
apodization function, P (x, y), that is positive in a continuous region conned to the
half plane where x > 0 and that is symmetrical about the x-axis, combined with the
reection of P around the y-axis multiplied by the phase factor e−i∆φ. A more general
case is illustrated in Figure 3.7.
Under these conditions the generalised pupil function, P, can be written as
1Solutions generally involve sine-integral, cosine-integral or exponential-integral terms.
42
x1
y1
e−i∆φ
Figure 3.7: Bilaterally Symmetric Segmented Pupil Mask
P(x, y) = P (x, y) + e−i∆φP (−x, y).
The complex, incoherent OTF is then computed as the normalised complex auto-
correlation (denoted ?) of the generalised pupil function. The correlation operation
can be changed into a convolution operation. Doing this is advantageous because the
convolution operation is commutative, associative and distributive whereas correlation
is not generally commutative. The required identity [32] is
f(x, y) ? g(x, y) = f(−x,−y)⊗ g∗(x, y).
The superscript ∗ denotes complex conjugation. The autocorrelation of P is then
43
expanded and simplied using the y-symmetry of P . Therefore:
P(x, y) ?P(x, y) =[P (x, y) + e−i∆φP (−x, y)
]?[P (x, y) + e−i∆φP (−x, y)
]
= P (x, y) ? P (x, y) + P (x, y) ?[e−i∆φP (−x, y)
]
+[e−i∆φP (−x, y)
]? P (x, y) +
[e−i∆φP (−x, y)
]?[e−i∆φP (−x, y)
]
= P (−x, y)⊗ P (x, y) + P (−x, y)⊗[ei∆φP (−x, y)
]
+[e−i∆φP (x, y)
]⊗ P (x, y) +
[e−i∆φP (x, y)
]⊗[ei∆φP (−x, y)
]
= P (−x, y)⊗ P (x, y) + ei∆φP (−x, y)⊗ P (−x, y)
+ e−i∆φP (x, y)⊗ P (x, y) + P (x, y)⊗ P (−x, y)
= 2P (x, y)⊗ P (−x, y) + ei∆φP (−x, y)⊗ P (−x, y)
+ e−i∆φP (x, y)⊗ P (x, y). (3.14)
The incoherent PSF is proportional to the Fourier Transform of the incoherent OTF
and the OTF is the autocorrelation of the pupil function (see Figure 2.2). The Fourier
Transform of the autocorrelation of the generalised pupil function P therefore provides
the PSF as:
F [P(x, y) ?P(x, y)] ∝ 2F [P (x, y)⊗ P (−x, y)] + ei∆φF [P (−x, y)⊗ P (−x, y)]
+ e−i∆φF [P (x, y)⊗ P (x, y)]
= 2F [P (x, y)]F [P (−x, y)]
+ ei∆φ F [P (−x, y)]2 + e−i∆φ F [P (x, y)]2 . (3.15)
As a rst check on the result in Equation 3.15, the rectangular aperture arrangement
described by Equation 3.3 and illustrated in Figure 3.3 will be used and the result
compared to Equation 3.12. The equivalent pupil apodization function can be written
44
as:
P (x1, y1) = rect
(x1 − x0
Dx
)rect
(y1
Dy
).
The Fourier transform of this function is,
Fω [P (x1, y1)] = Dxsinc(Dxωx)e−i2πωxx0Dysinc(Dyωy).
The Fourier transform of the reection across the y-axis is
Fω [P (−x1, y1)] = Dxsinc(Dxωx)ei2πωxx0Dysinc(Dyωy).
Thus
Fω [P(x1, y1) ?P(x1, y1)] ∝ 2 [Dxsinc(Dxωx)Dysinc(Dyωy)]2
+ ei∆φei4πωxx0 [Dxsinc(Dxωx)Dysinc(Dyωy)]2
+ e−i∆φe−i4πωxx0 [Dxsinc(Dxωx)Dysinc(Dyωy)]2
= [Dxsinc(Dxωx)Dysinc(Dyωy)]2
×
2 + ei4πωxx0ei∆φ + e−i4πωxx0e−i∆φ
= 2 [Dxsinc(Dxωx)Dysinc(Dyωy)]2
× 1 + cos (4πωxx0 + ∆φ) .
With ωx = x2
λfland ωy = y2
λfl, this is essentially the same result to within a scaling
constant and the sign of ∆φ as that obtained in Equation 3.12. This helps to establish
the validity of both Equation 3.15 and Equation 3.12.
45
x1
y1
e−i∆φ
Figure 3.8: Fractional Hilbert Phase Mask
3.5 Circular Pupils
For general imaging systems, the most common pupil shape is circular. Therefore a fur-
ther interesting case is that of a circular pupil, segmented into two semi-circles, one of
which receives the phase shift e−i∆φ. This is more generally known as the fractional Hil-
bert phase mask, illustrated in Figure 3.8. A few alternative approaches were explored
to reach an analytical image plane irradiance distribution expression for the fractional
Hilbert pupil mask. Some of these methods involved separation of the Hilbert pupil
phase mask into various combinations of circ and step functions, or expressed as innite
Fourier summations.
All of these alternative were found to have signicant analytical barriers.
As an illustration, a semi-circular pupil function can be written as
P (x1, y1) = circ
(√x2
1 + y21
R
)step (x1) . (3.16)
46
The circ function for strictly positive radius r is expressed as follows [74]:
circ(r) =
1 r < 1
12
r = 1
0 r > 1
. (3.17)
The Heaviside step function is dened [32] as,
step(x) =
0 x < 0
12
x = 0
1 x > 0
. (3.18)
The Fourier transform of the pupil apodization function in Equation 3.16 is a con-
volution of the Fourier transforms of the circ and step functions. This arises from the
convolution theorem, which states that the Fourier transform of a functional product is
the convolution of the Fourier transforms of the factors [76]. The Fourier transform of
the circ function is the jinc function or Airy disc [25], which is a scaled Bessel function
of the rst kind and of rst order, denoted J1, divided by the radial coordinate as:
Fω[
circ
(√x2 + y2
R
)]= R2
J1
(2πR
√ω2x + ω2
y
)
R√ω2x + ω2
y
= R2jinc(
2R√ω2x + ω2
y
),
where R is the pupil radius. The jinc function is dened here as:
jinc(ωx, ωy) = 2J1
(π√ω2x + ω2
y
)√ω2x + ω2
y
.
The Fourier transform of the step function [32] is
47
Fω [step(x)] =1
i2πω+ δ(ω),
where δ denotes the Dirac-delta function [32].
Therefore we can write:
Fω [P (x, y)] = Fω[
circ
(√x2 + y2
R
)]⊗Fω [step(x)]
= Fω[
circ
(√x2 + y2
R
)]⊗[
1
i2πωx+ δ(ωx)
]
= Fω[
circ
(√x2 + y2
R
)]⊗[
1
i2πωx
]+ Fω
[circ
(√x2 + y2
R
)].
This Fourier transform can also be expressed as a denite integral in polar coordin-
ates with the standard coordinate transform from Cartesian to polar coordinates [32]
as:
Fω[P (r, θ)] =
R
0
π2ˆ
−π2
re−i2πrωr cos(θ−ωθ)dθdr.
The generalised pupil function, P, now including the phase factor e−i∆φ can be
written as a Fourier series in θ multiplied by the circ function so that [76]:
P(r, θ) = circ( rR
)1
2− 2
π
∞∑
n=1,3,5...
1
nsinnθ + e−i∆φ
[1
2+
2
π
∞∑
n=1,3,5...
1
nsinnθ
].
Performing the Fourier transform of this complex pupil function is analytically chal-
lenging and thus was not taken any further.
The approaches that were attempted in reaching an analytical solution for the frac-
48
tional Hilbert phase mask all make the assumption that the central cut through the
phase retarder (Figure 3.1) is innitely thin. In practice, it will not be possible to imple-
ment a perfect cut in the phase retarder, so in the interest of reducing stray light, the cut
in the phase retarder should be masked with a thin, rectangular obscuration. This is the
equivalent of slightly shifting the position of the step function in Equation 3.16 to the
right. This however only adds to the computational problems already encountered. An
alternative approach to a solution for the fractional Hilbert phase mask was therefore
sought and this was found in the form of the Nijboer-Zernike approach [28].
3.6 Imperfect Nulling
With perfect geometry and perfectly monochromatic and uniform illumination, the
axial image irradiance is theoretically zero (interference null) for symmetric pair pupil
segment arrangements with an optical phase dierence of exactly π radians. In a real
experiment, none of these conditions will be exactly achieved and this section will
mention some of the eects of imperfections on the depth of the null.
3.6.1 Polychromaticity
No light source is perfectly monochromatic and unless the phase retarder is perfectly
achromatised, the phase delay introduced by the retarder will not be exactly the same
for all wavelengths. This is an inevitable consequence of the fact that all conventional
optical media exhibit some dispersion [77], meaning that the refractive index varies with
wavelength. In the case of a complete system, the overall spectral weighting depends
on the spectral distribution of the light source, the spectral transmission of the optical
train as well as the spectral response of the detector. These spectrally active eects will
be rolled up into a function of wavelength called the system Spectral Response Function
49
(SRF), denoted S(λ). The polychromatic PSF is then computed as an integral over
wavelength of the monochromatic PSF weighted by the SRF.
In principle it is possible to achromatise the phase retarder or at least to reduce the
magnitude of the dependence of retardation on wavelength. The wavelength dependence
of phase retardation is expressed through the refractive index as shown in A.1.
3.6.2 Non-Uniform Illumination or Vignetting
Non-uniform illumination of the pupil or lens vignetting eectively introduces a spatial
amplitude apodization eect. This would have to be taken into consideration when
analysing the eectiveness and depth of the interference null. For example, this could
appear as the non-negligible lens vignetting function, V , from Equation 3.11. Evalu-
ation of non-uniform illumination or vignetting has to be performed numerically, with
software such as Zemax®. Simple functional forms of spatial apodization could however
be addressed analytically.
3.6.3 Geometry Errors
Just as non-uniform illumination will result in imperfect nulling due to the reective
symmetry of the apodization function P being broken, so too will implementation errors
in the eective shape or dimensions of the apertures. For example, if the edges of the
apertures were not perfectly straight or orthogonal, this would have an eect on the
PSF. Again, this will have to be analysed numerically.
3.6.4 Aberrations
It is not possible to construct optical systems that produce and propagate perfect
optical wavefronts. Optical components in the system including lenses, windows, lters
50
and even the phase retarder itself will introduce non-zero optical wavefront aberrations
(distortions from ideal wavefront shape). In this case the generalised pupil function
must include these wavefront aberrations as a varying phase term over the aperture.
Nijboer-Zernike theory is an attractive alternative for dealing with arbitrary aberrations
in the propagated wavefront and the eect of such aberrations on the PSF.
3.7 Summary
A closed form solution to the focal plane EM eld intensity (Equation 3.12) has been
developed in this chapter for a rectangular aperture, segmented pupil (Figure 3.3),
phase-shifting interferometer for a point source. In practice, convolutions of this solu-
tion would be required to take nite source and detector sizes into account. Closed
form solution of these convolutions was not pursued here and a numerical approach
would be the rst choice in further work on this problem.
Analysis of the fractional Hilbert mask is further complicated by the computational
barriers encountered. The above alternatives lead to a search for a completely dierent
approach in solving this problem. This was found in the form of the Nijboer-Zernike
theory which enables computation of the focal plane eld through expansion of the
pupil function in terms of Zernike polynomials [28]. Further, in this investigation, the
extended Nijboer-Zernike approach (ENZ) that was encountered [78] expands the scope
of problems that can be addressed using Zernike-basis expansion.
The analysis of the fractional Hilbert mask in the context of ENZ theory resulted in
a journal paper submission to Current Applied Physics entitled Zernike-basis expansion
of the fractional and radial Hilbert phase masks. The paper is reproduced in its entirety
in Chapter 6.
51
Chapter 4
Precision Imaging Systems
4.1 Introduction
Precision optical imaging systems are used in a very broad range of applications. Some
of the most demanding imaging applications are those in microscopy, astronomy, space
optics and lithography. Design and construction of objective lens assemblies for ul-
traviolet lithography [15, 1820] is a key technology in the manufacture of state-of-
the-art electronic Integrated Circuits (IC). To illustrate the scale and complexity of
these systems, a lithography objective lens system patent circa 1998 [79] is shown in
cross-section in Figure 4.1. The completed opto-mechanical assembly of the Zeiss®
Starlith 1000 lithography lens is shown in Figure 4.2, which operates at a wavelength
of λ = 248 nm [80].
The lithography lens objective system illustrated in Figure 4.1 comprises 59 re-
fractive lens surfaces. Surface quality and cleanliness must be controlled to very high
standards in order to minimise the eects of stray light.
A variety of phenomena can degrade the contrast or quality of the image produced
by a precision optical system. To illustrate the extensive range of these eects, the
52
Figure 4.1: Lithography Lens Optical Design Cross-Section, USP 5,969,803
53
Figure 4.2: Zeiss® Starlith 1000 Lithography Lens Opto-Mechanical Assembly, Cour-tesy Electronics Production and Test [80]
54
Figure 4.3: Moon Image with Low Stray Light (left) and Simulated Stray Light (right)
following sections provide an introductory discussion, with a broad distinction into
design-related, manufacture-related and use-related eects.
While there are a great number of potential degrading eects, the unifying theme
is that of absorption or directional scatter of light. While absorption only reduces the
available light for the application, undesirable scattered light reaching the image plane
will reduce image quality. The general term for such undesirable scattered light is stray
light [6]. Figure 4.3 is a simulation of the eect that stray light could have on a high
quality image.
4.2 Optical Design and Opto-mechanical Design
The optical design of an imaging system is the primary determinant of the limiting
image quality that a system can deliver [81]. In the regime of geometrical optics,
system performance is analysed by performing geometrical tracing of rays through the
optical system according to the laws of refraction (Snell's law) and reection [22]. Any
deviations from perfect intersection by the rays at a geometrical point in the image
55
plane are referred to as geometrical ray aberrations [82]. The chromatic aberrations
are those related to the fact that optical glasses have a dierent refractive index for
dierent wavelengths of light. This is also referred to as optical dispersion [77, 83].
Generally, glass components also exhibit a change in dimensions as well as refractive
index with temperature change [84]. Hence, glass must be selected in the design process
to compensate both for dispersion and other physical and chemical properties of the
glass [85, 86].
Opto-mechanical design is the process of designing a structure to provide static
or dynamic (e.g. zoom lens) positional location and support to the active optical
components (lenses, mirrors, prisms, lters etc.) of an optical system [87]. The manner
of support has an inuence on such things as the self-weight distortion of an optical
component under the eect of gravity. Components can shift as a result of temperature
uctuations as the mechanical components expand and contract. Stress can be induced
through the mechanical pinching of optical components leading to birefringence [88],
which gives rise to double images [89].
Opto-mechanical components provide limiting apertures and other support struc-
tures such as spiders and mirror cells that diract and absorb light passing through
the system. Light can scatter from mechanical components and reach the image plane,
reducing contrast. This problem is generally controlled using carefully formulated black
paints or coatings [35,90] and by including baes which block the light scattered from
mechanical parts [6].
All glass materials exhibit a residual surface reection known as the fresnel reection
[22]. These reections can give rise to ghost images which interfere with primary image
utility. Ghost images are controlled by manipulating the geometrical layout of optical
components, as well as the application of thin optical lms (anti-reection coatings).
These thin lm coatings can have other inuences, such as increasing random scatter [91]
56
and distortion of the component through increased stress.
With due consideration to the image quality demanded of a system as well as the
operating environment in which the system must maintain performance, the combined
optical and opto-mechanical designs must be congured to within manufacturing toler-
ance bands to meet these requirements [87].
4.3 Optical Manufacture
Raw materials used in the production of precision optical systems are produced by
industrial processes which cannot yield materials that are perfect in every respect. For
example, optical glasses cannot be reproduced to have a perfectly precise refractive
index or with perfect spatial homogeneity [77]. Spatial uctuations in refractive index
can occur on scales of a fraction of a millimetre (striae) to gradients in refractive
index across a large glass sample [92, 93]. Glass can contain chemical impurities and
inclusions (bubbles, crystals or solid foreign matter) which absorb or scatter light [94].
While glass for precision optical systems is generally subject to the process of annealing
[77] to reduce internal stresses, there will always be some residual internal stress after
annealing [88].
Manufacture of precision optical components from the raw materials is executed
using a very broad range of industrial processes from traditional grind and pitch-
lap polishing [66] to state-of-the-art techniques such as Magneto-Rheological Finishing
(MRF [95]). The physical and chemical characteristics of the raw material in conjunc-
tion with the selected manufacturing processes results in a component which deviates
from the perfect geometry and optical properties of the designed component. Resid-
ual surface roughness (imperfect polish) and aws such as scratches and digs as well
as sub-surface damage [96] such as micro-cracks will also cause undesirable scattering.
57
The residual surface shape errors are often classied as form (or gure), being the low
spatial frequencies, ripple (mid-spatial frequencies) and nish (roughness, high spatial
frequencies) [97].
System integration involves the construction of the optical sub-assemblies and the
system from the components. The methods, controls and processes used in system
assembly can aect the quality of the resulting system [87].
Control of contamination from dust or other undesirable substances such as depos-
ited airborne Volatile Organic Compounds (VOC) can be a very important aspect of
the manufacturing process [98,99], especially in aerospace applications [100].
These manufacturing errors and deciencies, both in optical and opto-mechanical
components and assemblies will generally result in degraded image quality.
4.4 Operational Environment
Transition of the optical system into the operational environment, as well as the eects
of the operational environment itself can cause degradation in imaging performance. An
example of a severe transition environment is the launch of a space optical system [101],
with extreme levels of acceleration, shock, vibration and temperature uctuation.
Once an optical system is operating in the intended environment there are various
mechanisms by which imaging performance can be degraded [102]. While optical glasses
are generally very stable materials, they are susceptible to damage and ageing eects
[77]. These eects would include chipping, scratching, devitrication, solarisation, hard
radiation damage, surface etching and water damage. The optical thin lm coatings
applied to lens and mirror surfaces may delaminate or disintegrate due to one of the
above processes. Environmental contaminants, such as ngerprints and dust can cause
etching of coatings and underlying glass surfaces.
58
Solarisation is generally reversible damage to the spectral transmission of a trans-
parent material resulting from intense exposure to EM radiation (soft X-rays through
to ultraviolet wavelengths). Glasses used in space applications are usually subject to
higher levels of ultraviolet and shorter wavelength radiation in general.
If glass is subject to high uxes of EM radiation such as from a laser, specic types of
surface and bulk damage become a risk [103]. Existing damage as well as contamination
tend to increase the risk of further damage.
Hard radiation (e.g. protons, gamma rays) such as may be encountered in a space
environment can cause damage to glass types not formulated to be resistant to such
exposure [104,105].
Temperature uctuations not only cause variations in refractive index, but non-
uniform temperature distributions can cause component deformation and additional
stress-induced birefringence [88].
While the above eects relate to optical scatter from features occurring at various
physical size scales, it is important to note that there is also background scatter caused
at the fundamental physical granularity (atoms, molecules) of the materials comprising
the optical system. Elastic atomic and molecular scatter is also referred to as Rayleigh
scattering [5, 45, 106], which is relatively isotropic (the same over all scatter angles).
In particular, the gases between optical components as well as the glass materials will
exhibit both elastic and inelastic molecular scatter within the bulk of the material.
While Rayleigh scattering is usually at a substantially lower level of intensity than the
larger scale mechanisms discussed above, inelastic or non-linear scatter such as Raman
[45, 107, 108] scatter is usually of substantially lower intensity again than Rayleigh
scatter.
Little can be done to reduce molecular scatter. The optical system can be evacu-
ated to eliminate scattering from gases (normally the default situation in space optical
59
systems). This has to be taken into account during the design phase. In the case of
EUV lithographic systems, evacuation of the system is mandatory because air strongly
absorbs EM radiation at the operating wavelength of λ = 13.5 nm.
4.5 Summary
Optical imaging systems are used in a large variety of applications, some of which
place extreme demands on image quality. DUV/EUV lithography systems and space
telescopes are prominent examples. Optical systems in this class are vulnerable at every
stage of design, construction and deployment to factors which can degrade the quality
and utility of the images they produce. Some of these factors have been explored in this
chapter and clearly highlight the need to test and guarantee the quality of materials,
components, sub-assemblies and systems at every possible opportunity. This applies
equally to the procedures and processes used to create these systems as well as the
manufacturing and operational environments. Directional scattering of light plays a
central role and is the unifying principle across all of these considerations. Theoretical
understanding, modelling and measurement of optical scattering at material, component
and systems level are all mandatory in order to realise the full potential of these systems.
60
Chapter 5
Conclusion
This work undertook to review the applications, measurement and modelling of the
directional scatter of light, with an emphasis on precision optical imaging systems. Dif-
fraction, reection and refraction were identied as specic types of coherent scattering
along with random scatter due to optical surface defects, residual roughness, sub-surface
damage, coating structure and contamination amongst others.
The problem of stray light in high performance imaging systems was investigated
and it was found that by using a combination of the Generalised Harvey-Shack scatter
theory with the K-correlation scatter model, it is possible to calculate a surface BSDF
from prole measurements that is valid over a broad range of optical wavelengths,
incident and scattering angles as well as roughness levels. These models are suitable
for use in the design and analysis of stray light eects in precision imaging systems.
It was further determined that scatterometer design for measurement of weak scatter
as well as construction and operation of such scatterometers is inherently challenging. A
concept using segmented pupil interferometry (for example the fractional Hilbert phase
mask) has been proposed to cause a dark interference fringe to traverse the specular
beam, facilitating measurement of weak incoherent scatter near or within the specular
61
beam.
A large variety of possible deleterious eects related to design, manufacture and
use of imaging systems was observed. These eects range from design aws to optical
surface contamination. The unifying theme for all of these eects is the coherent or
random directional scattering of light.
Furthermore, it was found that techniques based on the ENZ approach oer consid-
erable promise for the purpose of analysing measurements of the system PSF in order
to quantify (retrieve) the various aberrations present in the imaging system.
A fractional Hilbert phase mask has been proposed and analysed as a means to
facilitate measurement of weak scatter within or near the specular beam. The mask
can also serve to enhance the ENZ aberration retrieval process by means of improving
PSF measurement diversity. These considerations lead to a paper entitled Zernike-
basis expansion of the fractional and radial Hilbert phase masks, which is provided in
Chapter 6.
62
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76
Chapter 6
Journal Paper Submission
The paper submitted to an accredited journal for publication is reproduced below. The
journal in question is Current Applied Physics, an Elsevier publication. The subject
matter of the paper arose from consideration of how to model image formation in the
presence of a phase retarder implemented as a tilted plano-parallel glass plate in the
pupil of an imaging system. This phase retarder was introduced in Chapter 3 as a
type of segmented pupil interferometer that could be of assistance when measuring
weak optical scatter near the specular beam. The technique could also be of use in
performing retrieval of optical phase or aberration information when evaluating the
image plane performance of optical systems. Considered in other contexts and more
generally, this type of circular-pupil phase retarder is better known as the fractional
Hilbert mask.
The acceptance correspondence and Article in Press are provided in Appendix B.
77
Zernike-basis expansion of the fractional and radial Hilbertphase masks
N Chetty1 and D J Griffith1
1School of Chemistry and Physics, University of KwaZulu-Natal (PMB)
Private Bag X01, Scottsville 3209, South Africa∗
compiled: October 2, 2014
The linear Hilbert phase mask or transform has found applications in image processing and spectroscopy. Anoptical version of the fractional Hilbert mask is considered here, comprising an imaging system with a circular,unobscured pupil in which a variable phase delay is introduced into one half of the pupil, split bilaterally.The radial Hilbert phase mask is also used in image processing and to produce optical vortices which haveapplications in optical tweezers and the detection of exoplanets.We subjected the fractional and radial Hilbert phase masks to Zernike function expansion in order to computethe image plane electromagnetic field distribution using Nijboer-Zernike theory. The Zernike functions forman orthogonal basis on the unit circle. The complex-valued Zernike expansion coefficients for these two phasemasks were derived for use in the context of the Extended Nijboer-Zernike (ENZ) theory of image formation.The ENZ approach is of interest in that it allows a greater range of defocus to be dealt with, provides a simplemeans of taking a finite source size into account and has been adapted to high Numerical Aperture (NA)imaging applications.Our image plane results for the fractional Hilbert mask were verified against a numerical model implementedin the commercial optical design and analysis code, Zemax®. It was found that the Nijboer-Zernike resultconverged to the Zemax® result from below as the number of Zernike terms in the expansion was increased.
1. Introduction
Many problems in optics relate to the diffraction prop-agation of electro-magnetic fields. In the modeling ofimaging systems, computation of the shape of the wave-front emerging from the system exit pupil is often fairlystraightforward and can be accomplished to adequateaccuracy using simple geometrical raytracing. The fi-nal step, that of diffraction propagation of the emergentwavefront into the focal region is then the key computa-tion in arriving at the complex field amplitude and hencethe radiant intensity at or near focus. This involves eval-uation of a diffraction integral, some of the most generalbeing the Rayleigh-Sommerfeld integrals [1].
This technique of using geometrical optics to map thefield in the exit pupil followed by numerical evaluationof a diffraction integral is one of the methods used in op-tical design and analysis software codes. The diffractionintegral that is numerically evaluated can be selected tosuit the problem. Various approximations can be usedto simplify the most general diffraction integrals, includ-ing the Debye, Fresnel and Fraunhofer approximations[1]. The Fresnel and Fraunhofer approximations are ofspecial significance in that diffraction propagation canbe expressed in terms of linear and shift-invariant oper-ations in frequency space giving rise to the domain ofFourier optics [2]. This permits application of all the
analytical and numerical Fourier techniques to a broadrange of optical diffraction problems.
Wavefront aberrations can be expanded in terms ofthe Zernike functions, which form an orthogonal set ofbasis functions on the unit circle. This covers the caseof the unobscured, circular pupil which is very commonin imaging systems. Not only does Zernike-basis expan-sion aid in the diagnostic interpretation of test interfer-ograms, for example, it also allows for application of theClassical Nijboer-Zernike (CNZ) theory in optical analy-sis [1]. In CNZ theory, each Zernike function mode in theexit pupil gives rise to a proportional field contributionin the image plane [1].
More recently [3], the Extended Nijboer-Zernike(ENZ) theory has been introduced permitting coverageof a broader range of diffraction imaging phenomena.A detailed description of the ENZ approach with ex-tensive references has been made available in [4]. Here,we apply the ENZ theory to analyse the circular pupil,fractional Hilbert mask [5] and perform a numerical veri-fication of the result. ENZ expansion coefficients for theradial Hilbert phase mask on a circular pupil are alsoprovided together with some illustrations of the PointSpread Function (PSF) in the focal region.
2. The Zernike Functions
The normalized Zernike functions Zmn (ρ) are composedof the product of a normalization factor Nm
n , a radialZernike polynomial Rmn (ρ) and azimuthal sine or cosinefactors for n ≥ 0 and m = −n,−n+ 2, . . . , n−2, n as [6]
2
Zmn (ρ, θ) =
Nmn R
mn (ρ) cosmθ m ≥ 0
−Nmn R
|m|n (ρ) sinmθ m < 0.
(1)
The restriction on m to vary from −n to n in incre-ments of 2 implies that if n is odd, m is also odd andif n is even, m will be even or zero. Conversely, if m isodd then n is also odd and if m is even or zero then nis also even or zero.
Normalization of the Zernike functions is not alwaysincluded in the definition (Equation 1) by authors (ex-cluded by Born and Wolf [1], included by Noll [7] andThibos et al. [6] for example). If the normalization fac-tors are excluded from the Zernike function definition,the normalization factors are then effectively incorpo-rated into the expansion coefficients (denoted αmn in thefollowing section).
While the product of the radial polynomial and co-sine/sine factors always varies from −1 to +1 [8, 9], thenormalization factor Nm
n for a specific Zernike functionis given by [6, 7]
Nmn =
√2(n+ 1)
1 + δm,0, (2)
where the Kronecker delta δm,n is
δm,n =
1 m = n
0 m 6= n.(3)
The radial Zernike polynomials can be written explic-itly [1] as
Rmn (ρ) =
p∑
s=0
(−1)s(n− s)!s!(p− s)!(q − s)!ρ
n−2s (4)
where p ≡ (n−|m|)/2 and q ≡ (n+|m|)/2 are integers thatare always zero or positive and the exclamation mark (!)denotes the factorial of the preceding expression. Theradial polynomial coefficients are all integers which al-ways sum to unity (i.e. Rmn (1) = 1 for all m and n).
2.A. Zernike Function OrthogonalityThe Zernike functions as defined in Equation 1 obey a setof inner product orthogonality relations [1]. The innerproduct of a function Φ(ρ, θ) with the Zernike functionsis defined as
〈Φ(ρ, θ), Zmn (ρ, θ)〉 ≡ 1
π
ˆ 2π
0
ˆ 1
0
Φ(ρ, θ)Zmn (ρ, θ)ρdρdθ.
(5)The inner products of the Zernike functions with them-selves are
〈Zmn (ρ, θ), Zm′
n′ (ρ, θ)〉 = δm,m′δn,n′ . (6)
For the radial polynomials
ˆ 1
0
Rmn (ρ)Rmn′(ρ)ρdρ =δn,n′
2(n+ 1), (7)
and
ˆ 2π
0
cosmθ cosm′θdθ = π(1 + δm,0)δm,m′ (8)
in the azimuthal direction. A similar relationship existsfor the sine functions. Orthonormality implies that theintegral of the product of any two Zernike functions overthe unit circle is zero unless the two functions are iden-tical (have the same n and m). In the latter case, theintegral is unity.
Since the Zernike functions form a complete, orthogo-nal basis on the unit circle, any (pupil) function Φ(ρ, θ)defined on the unit circle can be expanded in terms ofthe Zernike functions. The normalized Zernike coeffi-cients αmn in the Zernike representation of a function areunique for for a given pupil function, regardless of howmany terms are fitted. It also means that the coefficients(normalized in the following instance) for reconstructingΦ(ρ, θ) can be found using the inner product as
αmn = 〈Φ(ρ, θ), Zmn (ρ, θ)〉. (9)
If the Zernike functions are not normalized by defini-tion, the orthogonality relationship in Equation 6 willbe
〈Zmn (ρ, θ), Zm′
n′ (ρ, θ)〉 =1 + δm,02(n+ 1)
δm,m′δn,n′ (10)
=1
Nmn N
m′n′δm,m′δn,n′ . (11)
2.B. Properties and Relationships of the Zernike Ra-dial PolynomialsThe Zernike functions and radial polynomials have beenstudied in detail [9, 10] in the general context of orthog-onal polynomials [11]. Some special values and identitiesof the Zernike radial polynomials that will be requiredare [10]
Rmn (1) = 1, (12)
Rmn (0) = (−δm,0)n/2, (13)
ˆ 1
0
Rmn (ρ)dρ =(−1)p
n+ 1, (14)
ˆ 1
0
Rmn (ρ)ρdρ =
12 m,n = 0(−1)pmn(n+2) otherwise.
(15)
3
Equations 14 and 15 could not be found explicitly inexisting literature and are therefore proven here. Start-ing with a Zernike polynomial relationship provided byPrata and Rusch [12] and having defined
Smn (s, ρ) ≡ˆ
Rmn (ρ)ρsdρ, (16)
they provide
Smn (0, ρ) =1
n+ 1[(Rm+1
n+1 (ρ)−Rm+3n+1 (ρ) + . . .)
− (Rm+1n−1 (ρ)−Rm+3
n−1 (ρ) + . . .)], (17)
where the summations stop when the azimuthal (upper)index exceeds the radial index. This is now rewrittenusing summation notation as
Smn (0, ρ) =1
n+ 1
n+1∑
k=m+1,m+3...
(−1)k−m−1
2 Rkn+1(ρ)
− 1
n+ 1
n−1∑
k=m+1,m+3...
(−1)k−m−1
2 Rkn−1(ρ).
(18)
From Equation 16, the definite integral
ˆ 1
0
Rmn (ρ)dρ = Smn (0, ρ)− Smn (0, 0) = Smn (0, 1), (19)
since from Equation 18 with Equation 13 it is clear thatSmn (0, 0) = 0. Then substituting ρ = 1 in Equation 18and using the identity in Equation 12 yields
Smn (0, 1) =1
n+ 1
n+1∑
k=m+1,m+3...
(−1)k−m−1
2
− 1
n+ 1
n−1∑
k=m+1,m+3...
(−1)k−m−1
2 . (20)
The summations cancel term for term, except for thefinal term with k = n+1 in the first summation, so that
Smn (0, 1) =(−1)
n−m2
n+ 1=
(−1)p
n+ 1, (21)
thus proving Equation 14.Prata and Rusch [12] show that
Smn (1, ρ) =1
2(n+ 1)[(n+m+ 2)Sm+1
n+1 (0, ρ)
+ (n−m)Sm+1n−1 (0, ρ)]. (22)
From Equation 16, the definite integral
ˆ 1
0
Rmn (ρ)ρdρ = Smn (1, ρ)− Smn (1, 0) = Smn (1, 1). (23)
By substituting ρ = 0 in Equation 22 it is clear thatSmn (1, 0) = 0. Then substituting ρ = 1 in Equation 22and using Equation 21,
Smn (1, 1) =1
2(n+ 1)[(n+m+ 2)
(−1)n−m
2
n+ 2
+ (n−m)(−1)
n−m−22
n] (24)
=1
2(n+ 1)[(n+m+ 2)
(−1)p
n+ 2
− (n−m)(−1)p
n] (25)
=(−1)p
2(n+ 1)
[n+m+ 2
n+ 2− n−m
n
](26)
=(−1)pm
n(n+ 2). (27)
The final step is some straightforward algebra, thusproving Equation 15, except for the case m = n = 0,follows easily from R0
0(ρ) = 1.
3. Fractional Hilbert Mask Expansion
The goal here is to find the ENZ coefficients of the one-dimensional, fractional Hilbert mask on a circular, un-obscured pupil. Optically, this is equivalent to splittingthe pupil bilaterally and introducing a variable phaseshift into one half.
The Zernike coefficients are computed using the in-ner product as given in Equation 9 and the inner prod-uct definition in Equation 5. The ENZ approach differswith reference to the classical Nijboer-Zernike (CNZ)approach in that the complex-valued EM pupil fieldamplitude is expanded instead of the real-valued wave-front error. In the ENZ case, the formalism is essen-tially the same as for the classical approach, except thatthe Zernike coefficients are now generally also complex-valued [4] and are usually denoted βmn . We will use Amnand Bmn for the real and complex-valued Zernike coeffi-cients for the circular pupil fractional Hilbert mask.
For a general, complex-valued pupil function P (ρ, θ)the normalized ENZ expansion is (using the inner prod-uct notation defined in Equation 5)
βmn = 〈P (ρ, θ), Zmn (ρ, θ)〉. (28)
Note that in what follows, the ENZ expansions willonly consider cosine terms (m ≥ 0), and therefore anymodulus signs around m will be dropped.
To perform the ENZ expansion of the circular frac-tional Hilbert mask, the circular pupil is first segmentedbilaterally by expressing the (real-valued) pupil trans-mission A(θ) in the azimuthal coordinate, θ, as an in-finite Fourier cosine series representing a square wave[13–15] as
4
A(θ)e−i∆φ1−A(θ)
Figure 1. Fractional Hilbert Mask Pupil Function
A(θ) =1
2+
2
π
∞∑
k=1
sin kπ2
kcos kθ (29)
=1
2+
2
π
∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ. (30)
A Fourier series of this kind, when truncated, exhibitsthe Gibbs phenomenon (ripple and overshoot near dis-continuities [14]) and requires many terms in the sum-mation for reasonable convergence. This aspect will dis-cussed in more detail in §3.A.
The real transmission function A(θ) is ideally unityin the right half-plane (x > 0) and zero in the left half-plane (x < 0). So for the Hilbert mask, the right half ofthe pupil is expressed in transmission as A(θ), and theleft half of the pupil is then 1−A(θ).
The full, complex-valued pupil with a phase retarda-tion of ∆φ in the right half is then written as
P (ρ, θ) = A(θ)e−i∆φ + 1−A(θ)
= (e−i∆φ − 1)A(θ) + 1. (31)
This is illustrated in Figure 1.The real-valued transmission function A(θ) is subject
to normalized (coefficients for normalized Zernike func-
tions are denoted Amn , Bmn , αmn etc.) Zernike expansionfirst as
Amn = 〈A(θ), Zmn (ρ, θ)〉
=1
2〈Z0
0 , Zmn (ρ, θ)〉
+2
π〈
∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ, Zmn (ρ, θ)〉
=1
2〈Z0
0 , Zmn (ρ, θ)〉
+2Nm
n
π〈
∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ,Rmn (ρ) cosmθ〉.
(32)
The final inner product in Equation 32 is evaluatedusing the definition in Equation 6 as
〈∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ,Rmn (ρ) cosmθ〉
=1
π
2πˆ
0
1ˆ
0
∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ
Rmn (ρ) cosmθρdρdθ
=1
π
2πˆ
0
∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ
cosmθdθ
1ˆ
0
Rmn (ρ)ρdρ
=1
π
∞∑
k=1,3,5...
(−1)k−1
2
k
2πˆ
0
cos kθ cosmθdθ
1ˆ
0
Rmn (ρ)ρdρ.
(33)
Applying the cos-function orthogonality relationshipgiven in Equation 8 as well as Equation 15 we noticethat all terms in the summation vanish except for whenk = m. Thus
〈∞∑
k=1,3,5...
(−1)k−1
2
kcos kθ,Rmn (ρ) cosmθ〉
=1
π
∞∑
k=1,3,5...
(−1)k−1
2
kπδk,m
1ˆ
0
Rmn (ρ)ρdρ
=
(−1)
m−12
m(−1)
n−m2 m
n(n+2) m,npositive, odd
0 otherwise
=
(−1)
n−12
n(n+2) m,npositive, odd
0 otherwise.
(34)
Substituting Equation 34 in Equation 32 gives
Amn =
12 m = n = 0
2Nmn (−1)n−1
2
πn(n+2) m,n odd
0 m,n even.
(35)
It is noted that these coefficients are independent ofm (the coefficients are the same across all values of mfor each n).
The fractional Hilbert mask pupil function P (ρ, θ) isthen expanded using the ENZ procedure as
Bmn = 〈P (ρ, θ), Zmn (ρ, θ)〉= 〈(e−i∆φ − 1)A(θ) + 1, Zmn (ρ, θ)〉= (e−i∆φ − 1)Amn + 〈Z0
0 , Zmn (ρ, θ)〉. (36)
The Bmn coefficients can thus be written down from
the Amn coefficients (Equation 35) as
5
Bmn =
(e−i∆φ+1)2 m = n = 0
2Nmn (e−i∆φ−1)(−1)n−1
2
πn(n+2) m,n odd
0 m,n even.
(37)
This solution for Bmn has the required property that
if ∆φ → 0 then Bmn → 0 for all positive m and n while
B00 → 1. These coefficients are also independent of m.
3.A. Gibbs PhenomenonFormulation of the fractional Hilbert mask pupil func-tion involved the use of a square wave, synthesized asa Fourier cosine sequence A(θ) given in Equation 29.This series has relatively poor convergence and exhibitsripple (or ringing) and overshoot artifacts near the dis-continuities, known as “Gibbs Phenomenon” [14] whenthe series is truncated in numerical work. It follows thatthese artifacts will be propagated by the numerical ENZprocess to the focal region results when the series is nec-essarily truncated.
The Gibbs phenomenon can be suppressed to a signif-icant extent (but not eliminated) using a filtering factorcalled the Lanczos σ-factor [14]. If the series is to betruncated at n = N , the Fourier coefficients are multi-plied by sinc nN where sinc(x) ≡ sinπx/πx and the sum-mation is performed up to n = N −1. So truncating theseries at n = N and including the Lanczos σ-factor, theBmn coefficients for normalized Zernike functions become
Bmn =
e−i∆φ+12 m = n = 0
2Nmn (e−i∆φ−1)(−1)n−1
2 sinc nNπn(n+2) m,n odd, n < N
0 otherwise.
(38)
The corresponding result for the Amn , being the nor-malized Zernike coefficients for A(θ) follows from Equa-tion 35 as
Amn =
12 m = n = 0
2Nmn (−1)n−1
2 sinc nNπn(n+2) m,n odd, n < N
0 otherwise.
(39)
As an illustration, A(θ) is plotted for a summationof 50 terms with and without the Lanczos σ-factor inFigure 2. Clearly the inclusion of the σ-factor helps tosuppress the Gibbs phenomenon. The first discontinuityis at θ = π/2.
With or without the Lanczos σ-factor for suppressionof the Gibbs phenomenon, it is necessary to compute theZernike radial polynomials to high order for numericalverification purposes. The coefficients (see Equation 4)are ratios of factorials which become large, computation-ally expensive and potentially inaccurate from n = 40onward [16]. Alternative techniques for computation of
0.95
1.00
1.05
A(θ)
0 0.2 0.4 0.6 0.8 12θπ
N = 50 Without Lanczos σ-factor
N = 50 Including Lanczos σ-factor
Figure 2. A(θ) Truncated at 50 Terms
the polynomial coefficients are then required. Janssenand Dirksen [16] and Vlcek and Sovka [17] have providedsolutions which exploit the relationship of the Zernikeradial polynomials to Chebyshev polynomials and themore general Jacobi polynomials.
4. ENZ Propagation from Pupil to Focal RegionThe essence of the ENZ approach [3] is that each Zerniketerm in the complex-valued pupil expansion correspondsto a field contribution in the focal region. The complex-valued, scalar field amplitude, U , in the focal region iswritten as
U(ρ′, θ′; df ) =∑
m,n
βmn Umn (ρ′, θ′; df ), (40)
where βmn Umn is the contribution from βmn Z
mn in the
exit pupil, the lateral coordinates in the focal region are(ρ′, θ′), with a relative defocus from the best focal planein the axial direction of df . Janssen [3] evaluated thefocal region contributions Umn for substantial defocus df(relative to CNZ), by deriving functions V mn (ρ′, df ) suchthat
Umn (ρ′, θ′; df ) = 2imV mn (ρ′, df ) cosmθ′, (41)
where the radial and defocus function V mn is an integralover Zernike polynomial, radially-weighted Bessel func-tions, Jm of the first kind and order m,
V mn (ρ′, df ) =
ˆ 1
0
expidfρ
2Rmn (ρ)Jm(2πρρ′)ρdρ.
(42)Note that the focal region coordinates and defocus
parameter df are not in absolute length units here, but
6
expressed relative to the Numerical Aperture (NA) ofimaging and the wavelength of light under consideration.Exit pupil and focal region coordinates are illustrated in§4.A.
Power-Bessel series expansion on V mn produces [3]
V mn (ρ′, df ) = eidf∞∑
l=1
(−2idf )l−1
p∑
j=0
vljJm+l+2j(2πρ
′)
l (2πρ′)l,
(43)in which p = (n−m)/2 (as before for positive m). The vljfactor is computed as
vlj = (−1)p (m+ l + 2j)
(m+ j + l − 1
l − 1
)(j + l − 1
l − 1
)
×(l − 1
p− j
)/
(q + l + j
l
), (44)
with q = (n+m)/2 (as before for positive m).For defocus parameter df = 0, the result of the Classi-
cal Nijboer-Zernike (CNZ) theory can be used in whichthe summation over l in Equation 43 is reduced to asingle Bessel term as [4]
V mn (ρ′, 0) =
ˆ 1
0
Rmn (ρ)Jm(2πρρ′)ρdρ
=(−1)n−m
2Jn+1(2πρ′)
2πρ′. (45)
The CNZ result in Equation 45 is useful for consis-tency checking and for the initial evaluation of conver-gence. Also, the evaluation of the vlj binomial factorsas given in Equation 44 can become numerically prob-lematic at high order.
4.A. Pupil and Focal Region CoordinatesThe radial coordinate ρ in the pupil is normalized rel-ative to the physical pupil radius. The Cartesian coor-dinates (x′, y′, df ) and polar coordinates (ρ′, θ′) in thefocal region are also normalized relative to absolute co-ordinates (x, y, z) as follows:
x′ = xNA
λ= ρ′ cos θ′
y′ = yNA
λ= ρ′ sin θ′
df = 2π
λz(1−
√1−NA2), (46)
where NA is the Numerical Aperture (NA) of imagingand λ is the light wavelength.
The relative coordinate scheme is illustrated in Figure3. Here the system exit pupil P with absolute outer edgeradius R and relative radial coordinate ρ, has a sphericalwavefront emerging from the exit pupil and convergingon the image plane origin at O.
Figure 3. Exit Pupil (P ) and Focal Region (O) Coordinates
4.B. Zernike Function Normalization in ENZ Prac-ticeBraat et al. [4] exclude the normalisation factor Nm
n intheir definition of the Zernike functions . In that case,the expressions for the unnormalized coefficients Amn andBmn for the fractional Hilbert mask (with reference toEquations 39 and 38, including the Lanczos σ-factor)will include the square of the Nm
n factors and
Amn =
12 m = n = 0
2[Nmn ]2(−1)n−1
2 sinc nNπn(n+2) m,n odd, n < N
0 otherwise,
(47)
with
Bmn =
e−i∆φ+12 m = n = 0
2[Nmn ]2(e−i∆φ−1)(−1)n−1
2 sinc nNπn(n+2) m,n odd, n < N
0 otherwise.
(48)
4.C. ENZ Scope and LimitationsThe scope and limitations of the application of ENZtools circa 2010 is covered in detail by Van Haver [18].The accuracy of this ENZ result relative to more accu-rate (numerical) Rayleigh integral computations is ex-pected to degrade in both the low and high NA regimes.For optical wavelengths on the order of 550 nm in thevisible spectrum, the applicable range is taken to be0.05 < NA < 0.6. Other limitations relating to vectorand associated polarization effects are assumed negligi-ble in the current context.
5. Numerical Verification5.A. CNZ ConvergenceThe convergence of the CNZ result for the circularpupil, fractional Hilbert mask (Equations 45, 41, 40 with
7
10−6
10−5
10−4
10−3
10−2
10−1
100
|U(x,0)|2
−10 −5 0 5 10
x [µm]
N = 21 N = 41N = 81 Zemax
∆φ = π
Figure 4. Circular Fractional Hilbert Pupil Mask,log10 Nor-malized Image PSF x-Cross Section (CNZ), ∆φ = π, NA =0.1, λ = 0.5µm, df = 0
complex-valued coefficients Bmn computed with Equa-tions 48 or 38) was tested with increasing N (the max-imum value of n). A cross-section of the PSF inten-sity (irradiance) along the x-axis was computed for asequence of increasing N with ∆φ = π, NA = 0.1 andλ = 0.5µm. The CNZ result without the normalizationfactor is plotted in Figure 4. This result is shown rel-ative to the Non-Sequential Component (NSC) modelresult from the optical analysis package Zemax®. TheZemax® result will be discussed in more detail in §6.
A horizontal line at a relative irradiance which is 4 or-ders of magnitude below peak has been plotted in Fig-ures 4. This is assumed to be measurable with highdynamic range Focal Plane Array (FPA) detectors.
The x-cross section convergence of the CNZ result rel-ative to NSC Zemax® results for other values of ∆φ areplotted in Figures 5, 6 and 7.
For ∆φ = 0 (Figure 5), there is only a singleCNZ term, providing the normal incoherent diffraction-limited PSF for an unobscured, circular pupil (the Airydisk).
For non-zero ∆φ convergence appears to be from be-low. The image-plane PSF for a number of differentvalues of ∆φ was computed using the CNZ result (Equa-tion 45, df = 0) and the log10 of the normalized ir-radiance (square modulus of the complex-valued scalarfield) is plotted in Figure 8 for NA = 0.1, λ = 0.5µmand N = 41.
The result for ∆φ = 0 in Figure 8 yields the expecteddiffraction-limited PSF for NA = 0.1, λ = 0.5µm, corre-sponding to m = n = 0 in Equation 45. The full, centraldark interference null is only formed close to ∆φ = π.
Braat et al. [4] provide Matlab® code for evalua-
10−6
10−5
10−4
10−3
10−2
10−1
100
|U(x,0)|2
−10 −5 0 5 10
x [µm]
N = 21 N = 41N = 81 Zemax
∆φ = 0
Figure 5. Circular Fractional Hilbert Pupil Mask,log10 Nor-malized Image PSF x-Cross Section (CNZ), ∆φ = 0, NA =0.1, λ = 0.5µm, df = 0
10−6
10−5
10−4
10−3
10−2
10−1
100|U
(x,0)|2
−10 −5 0 5 10
x [µm]
N = 21 N = 41N = 81 Zemax
∆φ = π2
Figure 6. Circular Fractional Hilbert Pupil Mask, log10 Nor-malized Image PSF x-Cross Section (CNZ), ∆φ = π/2, NA =0.1, λ = 0.5µm, df = 0
tion of the V mn functions, comprising implementation ofEquation 43. This code was used for the computationof the field in the focal plane for the circular pupil, frac-tional Hilbert mask. The result was verified to be thesame for the df = 0 case as for the CNZ result.
In the process it was noted that the radial coordinateinput to the Matlab® function for V mn must be scaledby 2π in addition to normalizing as per Equation 46.
8
10−6
10−5
10−4
10−3
10−2
10−1
100
|U(x,0)|2
−10 −5 0 5 10
x [µm]
N = 21 N = 41N = 81 Zemax
∆φ = 910π
Figure 7. Circular Fractional Hilbert Pupil Mask, log10
Normalized Image PSF x-Cross Section (CNZ), ∆φ =9π/10, NA = 0.1, λ = 0.5µm, df = 0
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
∆φ = 0
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
∆φ = 12π
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
∆φ = 910π
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
∆φ = π
Figure 8. Circular Fractional Hilbert Pupil Mask, log Nor-malized Image PSF Irradiance (CNZ), NA = 0.1, λ =0.5µm, df = 0, N = 41
6. Comparison to Zemax® Results
Zemax® is a multipurpose, optical design and analysissoftware code. Two methods of modeling the circular,fractional Hilbert mask problem were attempted. In thefirst method, the phase piston (∆φ) was introduced intothe optical layout using a Zemax® surface type calledthe “Zernike Standard Phase” surface. This surface di-
Figure 9. Tilted plane parallel glass plate phase retarder asfractional Hilbert mask implementation. The phase retarderwould be placed in the system pupil.
rectly introduces a phase advance/retardation into thewavefront through specification of the Zernike-basis co-efficients. For various reasons (more detail is given in§6.A), this method did not work well and was useful only
to help verify the Zernike coefficients Amn for normalizedZernike functions (Equation 39).
The second Zemax® modeling method entailed intro-duction of the phase piston ∆φ using a Non-SequentialComponent (NSC) model of a tilted glass plate phaseretarder as illustrated in Figure 9. It was necessary touse two pairs of plates in sequence, with one sequen-tial pair tilted in opposite directions in order to undothe small lateral beam displacement caused by the firsttilted plate.
Once the Zemax® NSC model of the plate retarderhad been set up and verified, the cross-section of thePSF was computed, using direct summation of Huygenswavelets. This is thought to be the more accurate (anddefinitely more time-consuming) method with respect tothe more routine FFT approach. Besides the Huygenswavelet summation and FFT methods, Zemax® also of-fers a Physical Optical Propagation (POP) method ofanalysis. The Zemax® result from the Huygens wavelet-summation method is shown as a reference in Figures 4,5, 6 and 7. The CNZ result does converge to the Zemaxresult, but quite a large number of terms (N > 80) arerequired for close agreement.
9
6.A. Zernike Functions in Zemax
Zemax® follows the Noll [7] indexing scheme for theZernike (“standard” as opposed to “fringe”) functionswhich maps Zmn → Zj according to sequence A176988in the On-Line Encyclopedia of Integer Sequences. Thismapping appears to lack the virtue of a simple formula,but follows the rule that odd j are assigned to values ofm < 0 (and even j to all m > 0) and smaller values of jare assigned to smaller values of |m|.
Zemax only allows up to j = 231 Noll terms in theZernike series. This corresponds to all terms up ton = 21. This is not really sufficient to allow for an ac-curate computation of the fractional Hilbert pupil masksituation using the Zernike Standard Phase surface inZemax. Zemax® also follows Noll [7] in that the Zernikefunctions include the normalization factor.
6.B. Computation of Circular Pupil, FractionalHilbert Mask PSF using Zemax®
The circular pupil, fractional Hilbert mask was modeledin Zemax® using a Non-Sequential Component (NSC)group and implemented as a tilted plate phase retarderillustrated in Figure 9. Zemax® currently offers threedifferent methods of computing the PSF, namely the di-rect summation of Huygens wavelets, an FFT techniqueand a Physical Optics Propagation (POP) algorithm.All three methods produced essentially the same resultsillustrated in Figures 4 to 8.
The Huygens PSF results from Zemax® are visu-ally indistinguishable from the results shown in Figure8. Note that the x-axis cross sections of the Zemax®
Huygens PSF image results are plotted as the referencecurves in Figures 4, 5, 6 and 7.
7. The Radial Hilbert Phase Mask
The radial Hilbert phase mask is equivalent to a phasevortex used in [19] and in vortex coronagraph conceptsfor exoplanet detection [20]. In this case the pupil phaseis written as
P (θ) = eilθ, (49)
where the integer vortex topological charge is l ≥ 1. Weperformed unnormalised Zernike-basis expansion using
the ENZ approach and for m ≥ 0 as
βmn =⟨eilθ, Zmn (ρ, θ)
⟩
=1
π
2πˆ
0
1ˆ
0
eilθZmn (ρ, θ)ρdρdθ
=1
π
2πˆ
0
1ˆ
0
eilθRmn (ρ) cosmθρdρdθ
=1
π
2πˆ
0
eilθ cosmθdθ
1ˆ
0
Rmn (ρ)ρdρ
=1
π
2πˆ
0
(cos lθ + i sin lθ) cosmθdθ
1ˆ
0
Rmn (ρ)ρdρ
=δl,m(−1)pm
n(n+ 2). (50)
The final step arises from the orthogonality relationsof the sin and cos functions and application of Equation15. For m < 0, a similar argument yields
βmn =−iδl,|m|(−1)p|m|
n(n+ 2)= −iβ−mn . (51)
The vortex charge l selects the corresponding az-imuthal orders |m| = l only, but otherwise there arenon-zero coefficients for all n. The ENZ focal region re-construction proceeds according to Equation 40. Figure10 shows the intensity and phase in focus (df = 0) andout of focus (df = 3 and df = 6 ).
8. Conclusion
The Zernike functions were reviewed as an orthogonalset of basis functions on the unit circle. The Zernikefunction basis has found numerous applications in op-tics and other disciplines such as image processing. Aselection of Zernike function and Zernike radial polyno-mial relations were presented in the context of the Clas-sical Nijboer-Zernike (CNZ) theory of optical aberra-tions. The more recent, and growing, Extended Nijboer-Zernike (ENZ) approach [3, 18] has expanded the rangeof optical diffraction problems that can be modeled andanalyzed using an efficient semi-analytical procedurerooted in the use of Zernike-basis expansion.
The linear, fractional Hilbert mask has found applica-tions in optics [5] and image processing [21]. The mainresult presented here was derivation of the ExtendedNijboer-Zernike expansion coefficients for the circularpupil, fractional Hilbert mask. The convergence of theresulting infinite series, when truncated, was improvedby introduction of the Lanczos σ-factor [14]. The ENZcoefficients for the circular pupil, fractional Hilbert maskwere verified using numerical techniques, especially thatof a comparison to results from the Zemax® optical de-sign and analysis code.
10
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
Intensity, df = 0
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
Phase, df = 0
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
Intensity, df = 3
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
Phase, df = 3
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
Intensity, df = 6
−12
−9
−6
−3
0
3
6
9
12
y[µm]
−10 −5 0 5 10
x[µm]
Phase, df = 6
Figure 10. Optical vortex focal region normalized linear in-tensity and phase for charge l = 1. Normalised intensityvaries from 0 (black) to 1 (white) and phase varies from 0(black) to 2π(white), NA = 0.1, λ = 0.5µm.
The circular pupil ENZ expansion coefficients for theradial Hilbert phase mask, which is identified with theoptical vortex, have also been provided together withillustrations of the focal region PSF intensity and phase.
Our future work in this area will relate to the use ofHilbert masks to produce axial irradiance nulls for highdynamic range measurements of scattering phase func-tions and also for aberration retrieval using irradiancemeasurements in the focal region.
References
[1] M. Born and E. Wolf, Principles of Optics (CambridgeUniversity Press, 1999), 7th ed.
[2] J. W. Goodman, Introduction to Fourier Optics(McGraw-Hill, 1968).
[3] A. J. E. M. Janssen, “Extended Nijboer-Zernike ap-proach for the computation of optical point-spread func-tions”, J. Opt. Soc. Am. A 19, 5, 849-857 (2002).
[4] J. J. M. Braat, P. Dirksen, S. van Haver and A. J. E. M.
Janssen, “Detailed description of the ENZ approach”,(2013), http://www.nijboerzernike.nl/.
[5] A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Frac-tional Hilbert transform,” Opt. Lett. 21, 281-283 (1996).
[6] L. Thibos, R. A. Applegate, J. T. Schwiegerling, andR. Webb, “Standards for reporting the optical aberra-tions of eyes,” in “Vision Science and its Applications,”(Optical Society of America, 2000), p. SuC1.
[7] R. J. Noll, “Zernike polynomials and atmospheric tur-bulence”, J. Opt. Soc. Am. 66, 207-211 (1976).
[8] A. J. Janssen, “New analytic results for the Zernikecircle polynomials from a basic result in the Nijboer-Zernike diffraction theory,” Journal of the European Op-tical Society - Rapid publications 6 (2011).
[9] G. Szego, Orthogonal Polynomials, Vol. 23 in AmericanMathematical Society colloquium publications (Ameri-can Mathematical Society, 1939).
[10] W. J. Tango, “The circle polynomials of Zernike andtheir application in optics,” Applied Physics 13, 327-332 (1977).
[11] M. Abramowitz and I. A. Stegun, eds., “Handbook ofMathematical Functions,” Vol. 55 of Applied Mathe-matics Series (NIST, 1964), 10th ed.
[12] J. Aluizio Prata and W. V. T. Rusch, “Algorithm forcomputation of Zernike polynomials expansion coeffi-cients,” Appl. Opt. 28, 749-754 (1989).
[13] J. P. Boyd, Chebyshev and Fourier SpectralMethods, (Dover Publications, 2001), 2nd ed.,http://www-personal.umich.edu/~jpboyd/aaabook_
9500may00.pdf.[14] A. Jerri, The Gibbs Phenomenon in Fourier Analysis,
Splines and Wavelet Approximations, Vol. 446 of Math-ematics and Its Applications (Springer, 1998).
[15] K. Howell, Principles of Fourier Analysis, Studies in Ad-vanced Mathematics (Taylor & Francis, 2010).
[16] A. Janssen and P. Dirksen, “Computing Zernike poly-nomials of arbitrary degree using the discrete Fouriertransform,” Journal of the European Optical Society -Rapid publications 2 (2007).
[17] M. Vlcek and P. Sovka, “Zernike polynomials and theirspectral representation,” in Proceedings of the 2013 In-ternational Conference on Electronics, Signal Processingand Communication Systems, (2013).
[18] S. Van Haver, “The Extended Nijboer-Zernike Diffrac-tion Theory and its Applications,” Ph.D. thesis (TUDelft, 2010), http://repository.tudelft.nl/view/
ir/uuid:8d96ba75-24da-4e31-a750-1bc348155061/.[19] D. G. Grier, “A revolution in optical manipulation,” Na-
ture 424, 810816 (2003).[20] G. Foo, D. M. Palacios, and J. Grover A. Swartzlander,
“Optical vortex coronagraph,” Opt. Lett. 30, 33083310(2005).
[21] J. A. Davis, D. E. McNamara, D. M. Cottrell, andJ. Campos, “Image processing with the radial Hilberttransform: theory and experiments,” Opt. Lett. 25, 99-101 (2000).
Appendix A
Derivations
A.1 Phase Delay of a Tilted Plane Parallel Plate
The following is a derivation of the optical wavefront retardation introduced by tilting
of a plane parallel glass plate in collimated light. Starting with Snell's Law [109]
expressed for the angles of incidence, θi and refraction, θr relative to the surface normal
n on entry to a glass plate of thickness t having refractive index nr embedded in a
medium of refractive index ni:
ni sin θi = nr sin θr. (A.1)
In this case, the rotation of the plate from normal incidence is equal to θi the angle
of incidence (see Figure A.1).
If d is the physical path length within the plate then:
sin θr =
√d2 − t2d
, (A.2)
so that,
88
n
θi
θrt
d
nr
ni
Figure A.1: Ray Deviation in a Tilted Plane Parallel Plate
ni sin θinr
=
√d2 − t2d
. (A.3)
Solving for d as a function of θi leads to
d(θi) =nrt√
n2r − n2
i sin2 θi. (A.4)
The Optical Path Length (OPL), denoted Λ within the plate is the product of the
refractive index of the plate and the physical path length as:
Λ(θi) = nrd =n2rt√
n2r − n2
i sin2 θi. (A.5)
The Optical Path Dierence (OPD), denoted ∆Λ between a ray passing through the
tilted plate and untilted plate is
∆Λ(θi) = nrd− nrt =n2rt√
n2r − n2
i sin2 θi− nrt. (A.6)
89
If instead of referencing the refractive indices to the speed of light in vacuo, they
are referenced to that of the incident medium (typically air, having a refractive index
very slightly above unity), then setting n = nr/ni the OPD reduces to:
∆Λ(θi) =n2t√
n2 − sin2 θi− nt = nt
(n√
n2 − sin2 θi− 1
). (A.7)
If the OPD is to be expressed as a phase retardation ∆φ in radians then the OPD
is divided by the optical wavelength and multiplied by 2π as:
∆φ(θi) =2π∆Λ(θi)
λ=
2πnt
λ
(n√
n2 − sin2 θi− 1
). (A.8)
The wavelength, λ is measured in the reference medium.
90
Appendix B
Article in Press and Acceptance
Correspondence
The email correspondence from the journal Current Applied Physics, as well as the
Article in Press are provided below.
91
(27/03/2015) Derek Griffith - FW: Article tracking [CAP_3911] - Accepted Page 1
From: Naven Chetty <[email protected]>To: "Derek Griffith ([email protected])" <[email protected]>Date: 27/03/2015 09:12Subject: FW: Article tracking [CAP_3911] - Accepted manuscript available online (unedited version)
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Article title: Zernike-basis expansion of the fractional and radial Hilbert phase masksReference:: CAP3911Journal title: Current Applied PhysicsCorresponding author: Dr. N ChettyFirst author: Dr. N ChettyAccepted manuscript (unedited version) available online: 21-MAR-2015 DOI information: 10.1016/j.cap.2015.03.017
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Zernike-basis expansion of the fractional and radial Hilbert phasemasksQ1,6
Q5 N. Chetty*, D.J. GriffithSchool of Chemistry and Physics, University of KwaZulu-Natal (PMB), Private Bag X01, Scottsville 3209, South Africa
a r t i c l e i n f o
Article history:Received 7 January 2015Accepted 20 March 2015Available online xxx
a b s t r a c t
The linear Hilbert phase mask or transform has found applications in image processing and spectroscopy.An optical version of the fractional Hilbert mask is considered here, comprising an imaging systemwith acircular, unobscured pupil in which a variable phase delay is introduced into one half of the pupil, splitbilaterally.
The radial Hilbert phase mask is also used in image processing and to produce optical vortices whichhave applications in optical tweezers and the detection of exoplanets.
We subjected the fractional and radial Hilbert phase masks to Zernike function expansion in order tocompute the image plane electromagnetic field distribution using Nijboer-Zernike theory. The Zernikefunctions form an orthogonal basis on the unit circle. The complex-valued Zernike expansion coefficientsfor these two phase masks were derived for use in the context of the Extended Nijboer-Zernike (ENZ)theory of image formation. The ENZ approach is of interest in that it allows a greater range of defocus tobe dealt with, provides a simple means of taking a finite source size into account and has been adapted tohigh Numerical Aperture (NA) imaging applications.
Our image plane results for the fractional Hilbert mask were verified against a numerical modelimplemented in the commercial optical design and analysis code, Zemax®. It was found that the Nijboer-Zernike result converged to the Zemax® result from below as the number of Zernike terms in theexpansion was increased.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
Many problems in optics relate to the diffraction propagation ofelectro-magnetic fields. In the modeling of imaging systems,computation of the shape of the wavefront emerging from thesystem exit pupil is often fairly straightforward and can beaccomplished to adequate accuracy using simple geometrical ray-tracing. The final step, that of diffraction propagation of the emer-gent wavefront into the focal region is then the key computation inarriving at the complex field amplitude and hence the radiant in-tensity at or near focus. This involves evaluation of a diffractionintegral, some of the most general being the Rayleigh-Sommerfeldintegrals [1].
This technique of using geometrical optics tomap the field in theexit pupil followed by numerical evaluation of a diffraction integralis one of the methods used in optical design and analysis software
codes. The diffraction integral that is numerically evaluated can beselected to suit the problem. Various approximations can be used tosimplify the most general diffraction integrals, including the Debye,Fresnel and Fraunhofer approximations [1]. The Fresnel andFraunhofer approximations are of special significance in thatdiffraction propagation can be expressed in terms of linear andshift-invariant operations in frequency space giving rise to thedomain of Fourier optics [2]. This permits application of all theanalytical and numerical Fourier techniques to a broad range ofoptical diffraction problems.
Wavefront aberrations can be expanded in terms of the Zernikefunctions, which form an orthogonal set of basis functions on theunit circle. This covers the case of the unobscured, circular pupilwhich is very common in imaging systems. Not only does Zernike-basis expansion aid in the diagnostic interpretation of test in-terferograms, for example, it also allows for application of theClassical Nijboer-Zernike (CNZ) theory in optical analysis [1]. In CNZtheory, each Zernike function mode in the exit pupil gives rise to aproportional field contribution in the image plane [1].* Corresponding author.
E-mail address: [email protected] (N. Chetty).
Contents lists available at ScienceDirect
Current Applied Physics
journal homepage: www.elsevier .com/locate/cap
http://dx.doi.org/10.1016/j.cap.2015.03.0171567-1739/© 2015 Elsevier B.V. All rights reserved.
Current Applied Physics xxx (2015) 1e9
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CAP3911_proof 23 March 2015 1/9
Please cite this article in press as: N. Chetty, D.J. Griffith, Zernike-basis expansion of the fractional and radial Hilbert phase masks, CurrentApplied Physics (2015), http://dx.doi.org/10.1016/j.cap.2015.03.017
More recently [3], the Extended Nijboer-Zernike (ENZ) theoryhas been introduced permitting coverage of a broader range ofdiffraction imaging phenomena. A detailed description of the ENZapproach with extensive references has been made available in [4].Here, we apply the ENZ theory to analyse the circular pupil, frac-tional Hilbert mask [5] and perform a numerical verification of theresult. ENZ expansion coefficients for the radial Hilbert phase maskon a circular pupil are also provided together with some illustra-tions of the Point Spread Function (PSF) in the focal region.
2. The Zernike functions
The normalized Zernike functions bZmn ðrÞ are composed of the
product of a normalization factor Nmn , a radial Zernike polynomial
Rmn ðrÞ and azimuthal sine or cosine factors for n 0 andm ¼ n,nþ2,…,n2,n as [6].
bZmn ðr; qÞ ¼
Nmn Rmn ðrÞcosmq m 0
Nmn Rjmj
n ðrÞsin mq m<0: (1)
The restriction on m to vary from n to n in increments of 2implies that if n is odd,m is also odd and if n is even,mwill be evenor zero. Conversely, ifm is odd then n is also odd and ifm is even orzero then n is also even or zero.
Normalization of the Zernike functions is not always included inthe definition (Equation (1)) by authors (excluded by Born andWolf[1], included by Noll [7] and Thibos et al. [6] for example). If thenormalization factors are excluded from the Zernike functiondefinition, the normalization factors are then effectively incorpo-rated into the expansion coefficients (denoted amn in the followingsection).
While the product of the radial polynomial and cosine/sinefactors always varies from 1 to þ1 [8,9], the normalization factorNmn for a specific Zernike function is given by [6,7].
Nmn ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðnþ 1Þ1þ dm;0
s; (2)
where the Kronecker delta dm,n isQ2
dm;n ¼1 m ¼ n0 msn
: (3)
The radial Zernike polynomials can be written explicitly [1] as
Rmn ðrÞ ¼Xps¼0
ð1Þsðn sÞ!s!ðp sÞ!ðq sÞ!r
n2s (4)
where p≡ðn jmjÞ=2 and q≡ðnþ jmjÞ=2 are integers that are al-ways zero or positive and the exclamation mark (!) denotes thefactorial of the preceding expression. The radial polynomial co-efficients are all integers which always sum to unity (i.e. Rmn ð1Þ ¼ 1for all m and n).
2.1. Zernike function orthogonality
The Zernike functions as defined in Equation (1) obey a set ofinner product orthogonality relations [1]. The inner product of afunction Fðr; qÞ with the Zernike functions is defined as
DFðr; qÞ; bZm
n ðr; qÞE≡1p
Z2p0
Z10
Fðr; qÞbZmn ðr; qÞrdrdq: (5)
The inner products of the Zernike functions with themselves are
DbZmn ðr; qÞ; bZm0
n0 ðr; qÞE¼ dm;m0dn;n0 : (6)
For the radial polynomials
Z10
Rmn ðrÞRmn0 ðrÞrdr ¼ dn;n0
2ðnþ 1Þ ; (7)
and
Z2p0
cosmqcosm0qdq ¼ p1þ dm;0
dm;m0 : (8)
in the azimuthal direction. A similar relationship exists for the sinefunctions. Orthonormality implies that the integral of the productof any two Zernike functions over the unit circle is zero unless thetwo functions are identical (have the same n and m). In the lattercase, the integral is unity.
Since the Zernike functions form a complete, orthogonal basison the unit circle, any (pupil) function Fðr; qÞ defined on the unitcircle can be expanded in terms of the Zernike functions. Thenormalized Zernike coefficients bam
n in the Zernike representation ofa function are unique for a given pupil function, regardless of howmany terms are fitted. It also means that the coefficients (normal-ized in the following instance) for reconstructing Fðr; qÞ can befound using the inner product as
bamn ¼
DFðr; qÞ; bZm
n ðr; qÞE: (9)
If the Zernike functions are not normalized by definition, theorthogonality relationship in Equation (6) will beDZmn ðr; qÞ; Zm0
n0 ðr; qÞE¼ 1þ dm;0
2ðnþ 1Þ dm;m0dn;n0 (10)
¼ 1Nmn Nm0
n0dm;m0dn;n0 : (11)
2.2. Properties and relationships of the Zernike radial polynomials
The Zernike functions and radial polynomials have been studiedin detail [9,10] in the general context of orthogonal polynomials[11]. Some special values and identities of the Zernike radial poly-nomials that will be required are [10].
Rmn ð1Þ ¼ 1; (12)
Rmn ð0Þ ¼dm;0
n=2; (13)
Z10
Rmn ðrÞdr ¼ ð1Þpnþ 1
; (14)
Z10
Rmn ðrÞrdr ¼
8>>><>>>:12
m;n ¼ 0
ð1Þpmnðnþ 2Þ otherwise
: (15)
Equations (14) and (15) could not be found explicitly in existing
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e92
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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130
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literature and are therefore proven here. Starting with a Zernikepolynomial relationship provided by Prata and Rusch [12] andhaving defined
Smn ðs; rÞ≡Z
Rmn ðrÞrsdr; (16)
they provide
Smn ð0; rÞ ¼1
nþ 1
hRmþ1nþ1 ðrÞ Rmþ3
nþ1 ðrÞ þ…
Rmþ1n1 ðrÞ
Rmþ3n1 ðrÞ þ…
i; (17)
where the summations stop when the azimuthal (upper) indexexceeds the radial index. This is now rewritten using summationnotation as
Smn ð0; rÞ ¼1
nþ 1
Xk¼mþ1;mþ3…
nþ1ð1Þkm1
2 Rknþ1ðrÞ
1nþ 1
Xk¼mþ1;mþ3…
n1ð1Þkm1
2 Rkn1ðrÞ: (18)
From Equation (16), the definite integral
Z10
Rmn ðrÞdr ¼ Smn ð0; rÞ Smn ð0;0Þ ¼ Smn ð0;1Þ; (19)
since from Equation (18) with Equation (13) it is clear thatSmn ð0;0Þ ¼ 0: Then substituting r¼1 in Equation (18) and using theidentity in Equation (12) yields
Smn ð0;1Þ ¼1
nþ 1
Xk¼mþ1;mþ3…
nþ1ð1Þkm1
2
1nþ 1
Xk¼mþ1;mþ3…
n1ð1Þkm1
2 : (20)
The summations cancel term for term, except for the final termwith k ¼ n þ 1 in the first summation, so that
Smn ð0;1Þ ¼ð1Þnm
2
nþ 1¼ ð1Þp
nþ 1; (21)
thus proving Equation (14).Prata and Rusch [12] show that
Smn ð1; rÞ ¼1
2ðnþ 1Þhðnþmþ 2ÞSmþ1
nþ1 ð0; rÞ
þ ðnmÞSmþ1n1 ð0; rÞ
i: (22)
From Equation (16), the definite integral
Z10
Rmn ðrÞrdr ¼ Smn ð1; rÞ Smn ð1;0Þ ¼ Smn ð1;1Þ: (23)
By substituting r¼0 in Equation (22) it is clear that Smn ð1;0Þ ¼ 0.Then substituting r¼1 in Equation (22) and using Equation (21),
Smn ð1;1Þ ¼1
2ðnþ 1Þ
"ðnþmþ 2Þ ð1Þnm
2
nþ 2þ ðnmÞ ð1Þnm2
2
n
#(24)
¼ 12ðnþ 1Þ
ðnþmþ 2Þ ð1Þp
nþ 2 ðnmÞ ð1Þp
n
(25)
¼ ð1Þp2ðnþ 1Þ
nþmþ 2
nþ 2 nm
n
(26)
¼ ð1Þpmnðnþ 2Þ : (27)
The final step is some straightforward algebra, thus provingEquation (15), except for the case m ¼ n ¼ 0, follows easily from.R00ðrÞ ¼ 1:
3. Fractional Hilbert mask expansion
The goal here is to find the ENZ coefficients of the one-dimensional, fractional Hilbert mask on a circular, unobscuredpupil. Optically, this is equivalent to splitting the pupil bilaterallyand introducing a variable phase shift into one half.
The Zernike coefficients are computed using the inner productas given in Equation (9) and the inner product definition in Equa-tion (5). The ENZ approach differs with reference to the classicalNijboer-Zernike (CNZ) approach in that the complex-valued EMpupil field amplitude is expanded instead of the real-valuedwavefront error. In the ENZ case, the formalism is essentially thesame as for the classical approach, except that the Zernike co-efficients are now generally also complex-valued [4] and are usu-ally denoted bmn . We will use Am
n and Bmn for the real and complex-valued Zernike coefficients for the circular pupil fractional Hilbertmask.
For a general, complex-valued pupil function P(r,q) thenormalized ENZ expansion is (using the inner product notationdefined in Equation (5))
bbmn ¼
DPðr; qÞ; bZm
n ðr; qÞE: (28)
Note that inwhat follows, the ENZ expansions will only considercosine terms (m0), and therefore any modulus signs around mwill be dropped.
To perform the ENZ expansion of the circular fractional Hilbertmask, the circular pupil is first segmented bilaterally by expressingthe (real-valued) pupil transmission A(q) in the azimuthal coordi-nate, q, as an infinite Fourier cosine series representing a squarewave [13e15] as
AðqÞ ¼ 12þ 2p
Xk¼1
∞ sin kp2
kcos kq (29)
¼ 12þ 2p
Xk¼1;3;5…
∞ ð1Þk12
kcos kq: (30)
A Fourier series of this kind, when truncated, exhibits the Gibbsphenomenon (ripple and overshoot near discontinuities [14]) andrequiresmany terms in the summation for reasonable convergence.This aspect will discussed in more detail in xIII A.
The real transmission function A(q) is ideally unity in the righthalf-plane (x>0) and zero in the left half-plane (x < 0). So for the
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e9 3
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66676869707172737475767778798081828384858687888990919293949596979899
100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130
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Hilbert mask, the right half of the pupil is expressed in transmissionas A(q), and the left half of the pupil is then 1A(q).
The full, complex-valued pupil with a phase retardation of Df inthe right half is then written as
Pðr; qÞ ¼ AðqÞeiDf þ 1 AðqÞ ¼eiDf 1
AðqÞ þ 1: (31)
This is illustrated in Fig. 1.The real-valued transmission function A(q) is subject to
normalized (coefficients for normalized Zernike functions aredenoted bAm
n , bBmn , bam
n etc.) Zernike expansion first as
bAmn ¼
DAðqÞ;bZm
n ðr;qÞE
¼12
DbZ00;bZmn ðr;qÞ
Eþ2p
* Xk¼1;3;5…
∞ ð1Þk12
kcoskq;bZm
n ðr;qÞ+
¼12
DbZ00;bZmn ðr;qÞ
Eþ2Nm
np
* Xk¼1;3;5…
∞ ð1Þk12
kcoskq;Rmn ðrÞcosmq
+:
(32)
The final inner product in Equation (32) is evaluated using thedefinition in Equation (6) as
* Pk¼1;3;5…
∞ ð1Þk12
kcos kq;Rmn ðrÞcosmq
+
¼ 1p
Z2p0
Z10
24 Xk¼1;3;5…
∞ ð1Þk12
kcos kq
35Rmn ðrÞcosmqrdrdq
¼ 1p
Z2p0
24 Xk¼1;3;5…
∞ ð1Þk12
kcos kq
35cosmqdqZ10
Rmn ðrÞrdr
¼ 1p
X∞k¼1;3;5…
ð1Þk12
k
Z2p0
cos kqcosmqdqZ10
Rmn ðrÞrdr:
(33)
Applying the cos-function orthogonality relationship given inEquation (8) as well as Equation (15) we notice that all terms in thesummation vanish except for when k ¼ m. Thus
* P∞k¼1;3;5…
ð1Þk12
kcos kq;Rmn ðrÞcosmq
+
¼ 1p
X∞k¼1;3;5…
ð1Þk12
kpdk;m
Z10
Rmn ðrÞrdr
¼
8>>>><>>>>:ð1Þm1
2
mð1Þnm
2 mnðnþ 2Þ m;n positive; odd
0 otherwise
¼
8>>>><>>>>:ð1Þn1
2
nðnþ 2Þ m;n positive; odd
0 otherwise:
(34)
Substituting Equation (34) in Equation (32) gives
bAmn ¼
8>>>>>>>><>>>>>>>>:
12
m ¼ n ¼ 0
2Nmn ð1Þn1
2
pnðnþ 2Þ m;n odd
0 m;n even:
(35)
It is noted that these coefficients are independent of m (thecoefficients are the same across all values of m for each n).
The fractional Hilbert mask pupil function P(r,q) is thenexpanded using the ENZ procedure as
bBmn ¼
DPðr; qÞ; bZm
n ðr; qÞE
¼D
eiDf 1AðqÞ þ 1; bZm
n ðr; qÞE
¼ eiDf 1
bAmn þ
DbZ00;bZmn ðr; qÞ
E:
(36)
The bBmn coefficients can thus be written down from the bAm
n co-efficients (Equation (35)) as
bBmn ¼
8>>>>>>>><>>>>>>>>:
eiDf þ 1
2
m ¼ n ¼ 0
2Nmn
eiDf 1
ð1Þn1
2
pnðnþ 2Þ m;n odd
0 m;n even:
(37)
This solution for bBmn has the required property that if Df/0
then bBmn /0 for all positivem and nwhile bB0
0/1. These coefficientsare also independent of m.
3.1. Gibbs phenomenon
Formulation of the fractional Hilbert mask pupil functioninvolved the use of a square wave, synthesized as a Fourier cosinesequence A(q) given in Equation (29). This series has relatively poorconvergence and exhibits ripple (or ringing) and overshoot artifactsnear the discontinuities, known as “Gibbs Phenomenon” [14] whenthe series is truncated in numerical work. It follows that these ar-tifacts will be propagated by the numerical ENZ process to the focalregion results when the series is necessarily truncated.
The Gibbs phenomenon can be suppressed to a significantextent (but not eliminated) using a filtering factor called theFig. 1. Fractional Hilbert mask pupil function.
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e94
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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130
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Lanczos s-factor [14]. If the series is to be truncated at n ¼ N, theFourier coefficients are multiplied by sinc n
N∞ wheresincðxÞ≡sinpx=px and the summation is performed up to n ¼ N1.So truncating the series at n¼ N and including the Lanczos s-factor,the bBm
n coefficients for normalized Zernike functions become
bBmn ¼
8>>>>>>>>><>>>>>>>>>:
eiDf þ 12
m ¼ n ¼ 0
2Nmn
eiDf 1
ð1Þn1
2 sin cnN
pnðnþ 2Þ m;n odd;n<N
0 otherwise:
(38)
The corresponding result for the bAmn , being the normalized
Zernike coefficients for A(q) follows from Equation (35) as
bAmn ¼
8>>>>>>>>><>>>>>>>>>:
12
m ¼ n ¼ 0
2Nmn ð1Þn1
2 sin cnN
pnðnþ 2Þ m;n odd;n<N
0 otherwise:
(39)
As an illustration, A(q) is plotted for a summation of 50 termswith andwithout the Lanczos s-factor in Fig. 2. Clearly the inclusionof the s-factor helps to suppress the Gibbs phenomenon. The firstdiscontinuity is at q ¼ p/2.
With or without the Lanczos s-factor for suppression of theGibbs phenomenon, it is necessary to compute the Zernike radialpolynomials to high order for numerical verification purposes. Thecoefficients (see Equation (4)) are ratios of factorials which becomelarge, computationally expensive and potentially inaccurate fromn ¼ 40 onward [16]. Alternative techniques for computation of the
polynomial coefficients are then required. Janssen and Dirksen [16]and Vlcek and Sovka [17] have provided solutions which exploit therelationship of the Zernike radial polynomials to Chebyshev poly-nomials and the more general Jacobi polynomials.
4. ENZ propagation from pupil to focal region
The essence of the ENZ approach [3] is that each Zernike term inthe complex-valued pupil expansion corresponds to a fieldcontribution in the focal region. The complex-valued, scalar fieldamplitude, U, in the focal region is written as
Ur0; q0; df
¼
Xm;n
bmn Umn
r0; q0;df
; (40)
where bmn Umn is the contribution from bmn Z
mn in the exit pupil, the
lateral coordinates in the focal region are ðr0; q0Þ, with a relativedefocus from the best focal plane in the axial direction of df. Janssen[3] evaluated the focal region contributions Um
n for substantialdefocus df (relative to CNZ), by deriving functions Vm
n ðr0; df Þ suchthat
Umn
r0; q0; df
¼ 2imVm
n
r0; df
cosmq0; (41)
where the radial and defocus function Vmn is an integral over Zer-
nike polynomial, radially-weighted Bessel functions, Jm of the firstkind and order m,
Vmn
r0; df
¼
Z10
expnidf r
2oRmn ðrÞJmð2prr0Þrdr: (42)
Note that the focal region coordinates and defocus parameter dfare not in absolute length units here, but expressed relative to theNumerical Aperture (NA) of imaging and the wavelength of lightunder consideration. Exit pupil and focal region coordinates areillustrated in xVI A.
Power-Bessel series expansion on Vmn produces [3].
Vmn
r0; df
¼ eidf
X∞l¼1
2idf
l1 Xpj¼0
vljJmþlþ2jð2pr0Þ
lð2pr0Þl; (43)
in which p ¼ (nm)/2 (as before for positive m). The vlj factor iscomputed as
vlj ¼ð1Þpðmþ lþ 2jÞmþ jþ l 1
l 1
jþ l 1l 1
l 1p j
qþ lþ j
l
;
(44)
with p ¼ (n þ m)/2 (as before for positive m).For defocus parameter df ¼ 0, the result of the Classical Nijboer-
Zernike (CNZ) theory can be used in which the summation over l inEquation (43) is reduced to a single Bessel term as [4].
Vmn ðr0;0Þ ¼
Z10
Rmn ðrÞJmð2prr0Þrdr ¼ ð1Þnm2Jnþ1ð2pr0Þ
2pr0: (45)
The CNZ result in Equation (45) is useful for consistencychecking and for the initial evaluation of convergence. Also, theevaluation of the vlj binomial factors as given in Equation (44) canFig. 2. A(q) truncated at 50 terms.
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e9 5
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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130
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become numerically problematic at high order.
4.1. Pupil and focal region coordinates
The radial coordinate r in the pupil is normalized relative to thephysical pupil radius. The Cartesian coordinates ðx0; y0; df Þ and polarcoordinates ðr0; q0Þ in the focal region are also normalized relative toabsolute coordinates (x,y,z) as follows:
x0 ¼ xNAl
¼ r0cos q0
y0 ¼ yNAl
¼ r0sin q0
df ¼ 2p
lz1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 NA2
p ;
(46)
where NA is the Numerical Aperture (NA) of imaging and l is thelight wavelength.
The relative coordinate scheme is illustrated in Fig. 3. Here thesystem exit pupil P with absolute outer edge radius R and relativeradial coordinate r, has a spherical wavefront emerging from theexit pupil and converging on the image plane origin at O.
4.2. Zernike function normalization in ENZ practice
Braat et al. [4] exclude the normalisation factor Nmn in their
definition of the Zernike functions. In that case, the expressions forthe unnormalized coefficients Am
n and Bmn for the fractional Hilbertmask (with reference to Equations (39) and (38), including theLanczos s-factor) will include the square of the Nm
n factors and
Amn ¼
8>>>>>>>>><>>>>>>>>>:
12
m ¼ n ¼ 0
2 Nmn2ð1Þn1
2 sin cnN
pnðnþ 2Þ m;n odd;n<N
0 otherwise;
(47)
with
Bmn ¼
8>>>>>>>>><>>>>>>>>>:
eiDf þ 12
m ¼ n ¼ 0
2 Nmn2eiDf 1
ð1Þn1
2 sin cnN
pnðnþ 2Þ m;n odd;n<N
0 otherwise:
(48)
4.3. ENZ scope and limitations
The scope and limitations of the application of ENZ tools circa2010 is covered in detail by Van Haver [18]. The accuracy of thisENZ result relative to more accurate (numerical) Rayleigh inte-gral computations is expected to degrade in both the low andhigh NA regimes. For optical wavelengths on the order of 550 nmin the visible spectrum, the applicable range is taken to be0.05 < NA < 0.6. Other limitations relating to vector and associ-ated polarization effects are assumed negligible in the currentcontext.
5. Numerical verification
5.1. CNZ convergence
The convergence of the CNZ result for the circular pupil, frac-tional Hilbert mask (Equations (40), (41) and (45) with complex-valued coefficients Bmn computed with Equation (48) or 38) wastested with increasing N (the maximum value of n). A cross-sectionof the PSF intensity (irradiance) along the x-axis was computed fora sequence of increasing N with Df ¼ p,NA ¼ 0.1 and l ¼ 0.5 mm.The CNZ result without the normalization factor is plotted in Fig. 4.This result is shown relative to the Non-Sequential Component(NSC) model result from the optical analysis package Zemax®. TheZemax® result will be discussed in more detail in xVI.
A horizontal line at a relative irradiance which is 4 orders ofmagnitude below peak has been plotted in Fig. 4. This is assumed to
Fig. 3. Exit pupil (P) and focal region (O) coordinates.Fig. 4. Circular Fractional Hilbert Pupil Mask, log10 Normalized Image PSF x-CrossSection (CNZ), Df ¼ p, NA ¼ 0.1, l ¼ 0.5 mm, df ¼ 0.
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e96
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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130
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be measurable with high dynamic range Focal Plane Array (FPA)detectors.
The x-cross section convergence of the CNZ result relative toNSC Zemax® results for other values of Df are plotted in Figs. 5e7.
For Df¼ 0 (Fig. 5), there is only a single CNZ term, providing thenormal incoherent diffraction-limited PSF for an unobscured, cir-cular pupil (the Airy disk).
For non-zero Df convergence appears to be from below. Theimage-plane PSF for a number of different values of Df wascomputed using the CNZ result (Equation (45), df¼0) and the log10of the normalized irradiance (square modulus of the complex-valued scalar field) is plotted in Fig. 8 for NA ¼ 0.1, l ¼ 0.5 mmand N ¼ 41.
The result for Df¼0 in Fig. 8 yields the expected diffraction-limited PSF for NA ¼ 0.1, l ¼ 0.5 mm, corresponding to m ¼ n ¼ 0in Equation (45). The full, central dark interference null is onlyformed close to Df ¼ p.
Braat et al. [4] provide Matlab® code for evaluation of the Vmn
functions, comprising implementation of Equation (43). This codewas used for the computation of the field in the focal plane for thecircular pupil, fractional Hilbert mask. The result was verified to bethe same for the df ¼ 0 case as for the CNZ result.
In the process it was noted that the radial coordinate input tothe Matlab® function for Vm
n must be scaled by 2p in addition tonormalizing as per Equation (46).
6. Comparison to Zemax® results
Zemax® is a multipurpose, optical design and analysis soft-ware code. Two methods of modeling the circular, fractionalHilbert mask problem were attempted. In the first method, thephase piston (Df) was introduced into the optical layout using aZemax® surface type called the “Zernike Standard Phase” surface.This surface directly introduces a phase advance/retardation intothe wavefront through specification of the Zernike-basis co-efficients. For various reasons (more detail is given in xVI A), thismethod did not work well and was useful only to help verify the
Zernike coefficients bAmn for normalized Zernike functions (Equa-
tion (39)).The second Zemax® modeling method entailed introduction of
the phase piston Df using a Non-Sequential Component (NSC)model of a tilted glass plate phase retarder as illustrated in Fig. 9. Itwas necessary to use two pairs of plates in sequence, with onesequential pair tilted in opposite directions in order to undo thesmall lateral beam displacement caused by the first tilted plate.Fig. 5. Circular Fractional Hilbert Pupil Mask, log10 Normalized Image PSF x-Cross
Section (CNZ), Df ¼ 0, NA ¼ 0.1, l ¼ 0.5 mm, df ¼ 0.
Fig. 6. Circular Fractional Hilbert Pupil Mask, log10 Normalized Image PSF x-CrossSection (CNZ).Df ¼ p
2; NA ¼ 0:1; l ¼ 0:5mm; df ¼ 0.
Fig. 7. Circular Fractional Hilbert Pupil Mask, log10 Normalized Image PSF x-CrossSection (CNZ).Df ¼ 9p
10; NA ¼ 0:1; l ¼ 0:5mm; df ¼ 0.
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e9 7
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66676869707172737475767778798081828384858687888990919293949596979899
100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130
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Once the Zemax® NSC model of the plate retarder had been setup and verified, the cross-section of the PSF was computed, usingdirect summation of Huygens wavelets. This is thought to be themore accurate (and definitely more time-consuming) method withrespect to the more routine FFT approach. Besides the Huygenswavelet summation and FFTmethods, Zemax® also offers a PhysicalOptical Propagation (POP) method of analysis. The Zemax® resultfrom the Huygens wavelet-summation method is shown as areference in Figs. 4e7. The CNZ result does converge to the Zemaxresult, but quite a large number of terms (N > 80) are required for
close agreement.
6.1. Zernike functions in Zemax
Zemax® follows the Noll [7] indexing scheme for the Zernike(“standard” as opposed to “fringe”) functions which maps Zmn /Zjaccording to sequence A176988 in the On-Line Encyclopedia ofInteger Sequences. This mapping appears to lack the virtue of asimple formula, but follows the rule that odd j are assigned tovalues of m < 0 (and even j to all m > 0) and smaller values of j areassigned to smaller values of jmj.
Zemax only allows up to j¼ 231 Noll terms in the Zernike series.This corresponds to all terms up to n ¼ 21. This is not really suffi-cient to allow for an accurate computation of the fractional Hilbertpupil mask situation using the Zernike Standard Phase surface inZemax. Zemax® also follows Noll [7] in that the Zernike functionsinclude the normalization factor.
6.2. Computation of circular pupil, fractional Hilbert mask PSFusing Zemax®
The circular pupil, fractional Hilbert mask was modeled inZemax® using a Non-Sequential Component (NSC) group andimplemented as a tilted plate phase retarder illustrated in Fig. 9.Zemax® currently offers three different methods of computing thePSF, namely the direct summation of Huygens wavelets, an FFTtechnique and a Physical Optics Propagation (POP) algorithm. Allthree methods produced essentially the same results illustrated inFigures 4 to 8.
The Huygens PSF results from Zemax® are visually indistin-guishable from the results shown in Fig. 8. Note that the x-axis crosssections of the Zemax® Huygens PSF image results are plotted asthe reference curves in Figs. 4e7.
7. The radial Hilbert phase mask
The radial Hilbert phase mask is equivalent to a phase vortexused in [19] and in vortex coronagraph concepts for exoplanetdetection [20]. In this case the pupil phase is written as
PðqÞ ¼ eilq; (49)
where the integer vortex topological charge is l 1. We performedunnormalised Zernike-basis expansion using the ENZ approach andfor m 0 as
bmn ¼Deilq; Zmn ðr; qÞ
E
¼ 1p
Z2p0
Z10
eilqZmn ðr; qÞrdrdq
¼ 1p
Z2p0
Z10
eilqRmn ðrÞcosmqrdrdq
¼ 1p
Z2p0
eilqcosmqdqZ10
Rmn ðrÞrdr
¼ 1p
Z2p0
ðcos lqþ i sin lqÞcosmqdqZ10
Rmn ðrÞrdr
¼ dl;mð1Þpmnðnþ 2Þ :
(50)
Fig. 9. Tilted plane parallel glass plate phase retarder as fractional Hilbert maskimplementation. The phase retarder would be placed in the system pupil.
Fig. 8. Circular Fractional Hilbert Pupil Mask, log Normalized Image PSF Irradiance(CNZ), NA ¼ 0.1, l ¼ 0.5 mm, df ¼ 0, N ¼ 41.
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The final step arises from the orthogonality relations of the sinand cos functions and application of Equation (15). For m < 0, asimilar argument yields
bmn ¼idl;jmjð1Þp
mnðnþ 2Þ ¼ ibm
n : (51)
The vortex charge l selects the corresponding azimuthal ordersjmj ¼ l only, but otherwise there are non-zero coefficients for all n.The ENZ focal region reconstruction proceeds according to Equa-tion (40). Fig. 10 shows the intensity and phase in focus (df ¼ 0) andout of focus (df ¼ 3 and df ¼ 6).
8. Conclusion
The Zernike functions were reviewed as an orthogonal set ofbasis functions on the unit circle. The Zernike function basis hasfound numerous applications in optics and other disciplines such asimage processing. A selection of Zernike function and Zernike radialpolynomial relations were presented in the context of the ClassicalNijboer-Zernike (CNZ) theory of optical aberrations. The more
recent, and growing, Extended Nijboer-Zernike (ENZ) approach[3,18] has expanded the range of optical diffraction problems thatcan be modeled and analyzed using an efficient semi-analyticalprocedure rooted in the use of Zernike-basis expansion.
The linear, fractional Hilbert mask has found applications inoptics [5] and image processing [21]. The main result presentedhere was derivation of the Extended Nijboer-Zernike expansioncoefficients for the circular pupil, fractional Hilbert mask. Theconvergence of the resulting infinite series, when truncated, wasimproved by introduction of the Lanczos s-factor [14]. The ENZcoefficients for the circular pupil, fractional Hilbert mask wereverified using numerical techniques, especially that of a compari-son to results from the Zemax® optical design and analysis code.
The circular pupil ENZ expansion coefficients for the radialHilbert phase mask, which is identified with the optical vortex,have also been provided together with illustrations of the focalregion PSF intensity and phase.
Our future work in this area will relate to the use of Hilbertmasks to produce axial irradiance nulls for high dynamic rangemeasurements of scattering phase functions and also for aberrationretrieval using irradiance measurements in the focal region.
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Fig. 10. Optical vortex focal region normalized linear intensity and phase for chargel ¼ 1. Normalised intensity varies from 0 (black) to 1 (white) and phase varies from0 (black) to 2p(white), NA ¼ 0.1, l ¼ 0.5 mm.
N. Chetty, D.J. Griffith / Current Applied Physics xxx (2015) 1e9 9
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100101102103104105106107108109110111112113114115116117118119120121122123124125126
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